Signature and Spectral Flow of $ J $-Unitary $ S^ 1$-Fredholm Operators

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Fehl if -Fredholm J T ) uiayoeaoscnb on in found be can operators -unitary ∗ ally satisfying naKenspace Krein a on t eoe h sa ibr space Hilbert usual the denotes oooyivratgiven invariant homotopy a y e ffiiemlilct)enjoys multiplicity) finite of ues S 1 gpe prtr a a has operators -gapped T J S ermany 2 1 − hs concepts These . = z 1 1 sFredholm is and ( K re J , J φ nley- ) σ ∗ = are ( e fun- led denotes T J ) ∗ In . fa of [GL] in the finite dimensional case. Even eigenvalue collisions stay on S1 whenever J is definite on the associated eigenspace. These results readily extend to the discrete spectrum in the infinite dimensional situation, and hold under certain further conditions also for essential spectrum (see Theorem 2.5.23 in [AI]). For the convenience of the reader and because it is relevant in the sequel, the main facts of stability theory for the discrete spectrum on S1 are reviewed in Section 3.4. This work is about Fredholm properties of J-unitary operators and homotopy invariants for associated operator classes. A J-unitary T is said to be S1-Fredholm if T z 1 is a Fredholm − operator for all z S1. It is said to be essentially S1-gapped if there is only discrete spectrum on S1. The associated∈ operator classes are denoted by F( ,J) and GU( ,J). Both are open subsets of the J-unitary operators. The essentially S1-gappedK J-unitaries formK a strict subset of the S1-Fredholm operators, as is shown by an explicit example in Section 6.4. This example also shows that the property to be essentially S1-gapped is not stable under compact (actually even finite dimensional) perturbations, while the S1-Fredholm property clearly is compactly stable. The first main topic of the paper is the signature Sig(T ) of an essentially S1-gapped J- unitary T . It is defined as the difference of the number of positive and negative eigenvalues of J restricted to the generalized eigenspace of all eigenvalues on S1 (see Section 5.2). The signature is a homotopy invariant (Theorem 8, based on Krein stability theory) which is trivial in finite dimension, but in infinite dimension it permits to split the set GU( ,J) of essentially S1-gapped J-unitaries into disjoint open components, similar as the FredholmK index does for the Fredholm operators. It is actually simple to produce examples with non-trivial signature, see Section 5.2, so that all components are non-empty. Furthermore, non-trivial signatures appear in applications. Section 6.8 consideres the J-unitary transfer operator of a half-space discrete magnetic Schödinger operator (Harper model). Its signature is then equal to the number of edge modes weighted by the sign of the group velocity, and this latter number is known to be equal to a Chern number of the planar Harper model. It is reasonable to expect that there is also a theory of essentially R-gapped J-self-adjoint operators and associated signature invariants. The second main topic concerns the homotopy theory of paths in F( ,J) of S1-Fredholm K operators. This uses the construction (Theorem 5) of a unitary operator V (T ) on which can be associated to every J-unitary operator T because its graph is Lagrangian. It is explicitlyK given by 1 1 a b (a∗)− bd− T = V (T ) = 1 1 . c d 7→ d− c d− − One then has the equivalence of Tφ = φ, φ , with V (T )φ = φ (Theorem 6). Furthermore, it turns out (Theorem 7) that for an S1-Fredholm∈ K operator T , the point 1 is not in the essential spectrum of V (T ). Hence for paths t T in the S1-Fredholm operators one can associate a 7→ t spectral flow of t V (Tt) through 1. Due to the above simple connection between the eigenvalue equations, this spectral7→ flow is an intersection index counting the weighted number of periodic solutions φt of Ttφt = φt along the path (see Section 4.2). Actually, this index can also be understood as a Bott-Maslov index [Bot, Mas], albeit in an infinite dimensional space. More precisely, the unitary V (Tt) describes a Lagrangian subspace given by a certain adequate graph of Tt, called the twisted graph. Therefore the above spectral flow can be considered as an infinite dimensional and modified version of the Conley-Zehnder index [CZ]. Because of this connection,

2 this paper also contains in Sections 2.3 and 2.4 a streamlined exposition of the Bott-Maslov index in infinite dimension. It is based on the use of unitaries to describe Lagrangian subspaces and the associated spectral flow similar as above. This approach does not use the Souriau map as prior works [Fur], or several charts as the approach in [GPP]. The Bott-Maslov index and the above intersection number (of Conley-Zehnder type) is of little use if one has no information about the orientation of intersections (or equivalently the orientation of the crossings of the spectral flow). These orientations have to be studied for every given concrete path. A standard example is a linear Hamiltonian system (here in infinite dimension) for which there is some monotonicity (Theorem 10) that is actually at the basis of Sturm-Liouville oscillation theory [Bot, Lid, SB1, SB2]. Another example of a monotonous path counts the bound states of Schrödinger equations (see Section 6.2). Furthermore, associated to 1 ıt every essentially S -gapped J-unitary T there is a path t [0, 2π) e− T in F( ,J). Its intersection number is equal to the signature Sig(T ) (Theorem∈ 9). This7→ particular pathK is not monotonous, but the orientation of every crossing is given by the inertia of the corresponding eigenvalue of T on S1. The author’s initial motivation to study S1-Fredholm operators roots in applications to solid state physics systems. Just as for the Harper model alluded to above, the half-space transfer operators of periodic systems at energies in a gap of the full-space operator are S1-Fredholm operators. Each eigenvalue on the unit circle then corresponds to an edge mode (a peculiarity is that a continuum of eigenvalues on S1 corresponds to a flat band of edge modes). These edge modes determine the boundary physics, and systems for which it is non-trivial are called topological insulators. Therefore the signature (and its variants for classes of operators with further symmetries) of associated half-space transfer operators allows to distinguish between different topological insulators and its homotopy invariance also shows their structural stability. This application will be explored in a subsequent publication [SV]. Finally a few words on the structure and style of the present paper. It is basically self- contained with detailed proofs except for a few functional analytical facts resembled in the appendices. Of course, this makes the paper lengthy. However, a large part of the text is anyway necessary in order to introduce notations. Moreover, the author hopes that some readers may appreciate the concise treatment of some already known results and that experts can easily localize the new results. Section 2 concerns the basic geometry of Krein spaces and introduces the Bott-Maslov index. Section 3 recalls some important properties of J-unitaries including Krein stability analysis, and provides a detailed analysis of the unitary V (T ) introduced above. Then Sections 4 and 5 study the S1-Fredholm operators and essentially S1-gapped operators respectively, as well as the intersection number and signature. Finally Section 6 provides examples to illustrate the general concepts of the paper. Notations: Vectors in Hilbert space are denoted by v,w and φ, ψ ; frames in (see ∈ H ∈ K K Appendix B for a definition) are denoted by Φ, Ψ; subspaces of (or in Appendix B) are described by , ; bounded operators on are described by smallK lettersH a,b,c,d B( ), while boundedE operatorsF on rather by capitalH letters T,S,K B( ); unitary operators∈ H are K ∈ K denoted by u U( ) and U, V U( ); the star is used for the passage to dual space in ∈ H ∈ K ∗ and (notably, φ∗ψ denotes the scalar product of φ, ψ ) as well as the Hilbert space K H ∈ K

3 adjoint operator; norms of vectors and the operator norm are written as . ; ﬁnally the essential k k spectrum σess(T ) is the spectrum of T without the discrete spectrum (thus in this work the essential spectrum is not given by those z for which T z1 is Fredholm). − 2 Geometry of subspaces of a Krein space

2.1 Isotropy and orthogonality in Krein space This section recalls a few deﬁnitions (e.g. [Bog, AI]), then provides some formulas for projections in terms of frames used later on.

Definition 1 Let be a subspace of a Krein space ( ,J). E K (i) is called positive definite for J if and only if φ∗Jφ > 0 for all non-zero φ . Similarly, E ∈E E is called negative definite for J if φ∗Jφ < 0 for all non-zero φ . Moreover, is called definite if it is either positive or negative definite. ∈E E

(ii) is called degenerate for J if and only if there is a non-zero vector φ such that φ∗Jψ =0 forE all ψ . If is not degenerate, it is called non-degenerate. ∈E ∈E E (iii) is called isotropic for J if and only if φ∗Jψ =0 for all φ, ψ . E ∈E (iv) A maximally isotropic subspace is called Lagrangian.

(v) is called coisotropic for J if and only if φ∗Jψ =0 for all ψ implies that φ . E ∈E ∈E Let us point out that the maximality condition implies that Lagrangian subspaces are always closed. Note also that deﬁnite subspaces are clearly non-degenerate, but the inverse is not true.

Deﬁnition 2 Let and ′ be two subspaces of a Krein space ( ,J). E E K (i) and ′ are called J-orthogonal if and only if φ∗Jψ =0 for all φ and ψ ′. We then E E ∈E ∈E also write ′. E⊥E b (ii) Given , its J-orthogonal complement ⊥ is given by all vectors φ satisfying φ∗Jψ =0 for all ψE b. E ∈K ∈E (iii) If and ′ are J-orthogonal and have trivial intersection, their sum is denoted by + ′. E E E E Just as in euclidean geometry, J-orthogonal complements are closed. The followingb lemma collects a few direct connections with the notions above.

Lemma 1 Let be a subspace of . b b E K (i) ( ⊥)⊥ = E E b (ii) is isotropic if and only if ⊥ E E⊂Eb (iii) is coisotropic if and only if ⊥ E E ⊂Eb (iv) is Lagrangian if and only if = ⊥ E E E b (v) is non-degenerate if and only if ⊥ = 0 E E∩E { }

4 b Other than in euclidean geometry, ⊥ and do not span , in general. But the following result shows that the dimensions add upE as usualE (a statementK that is not very interesting in inﬁnite dimension).

Proposition 1 Let and be subspaces of . Then: Eb bF K (i) ⊥ ⊥ E⊂Fb b ⇐⇒ F ⊂Eb b b b (ii) ⊥ + ⊥ =( )⊥ and ( + )⊥ = ⊥ ⊥ E F E∩Fb E F E ∩F (iii) Ker(J )= ⊥ |E E∩Eb (iv) dim( ) + dim( ⊥)=2n E E (v) Lagrangian subspaces of have dimension n. K

Proof. (i) to (iii) are obvious. (iv) For the usual orthogonal complement ⊥ to (w.r.t. to the E E euclidean scalar product on ), one has dim( ) + dim( ⊥)=2n. It is hence suﬃcient to provide K b E E b b an isomorphism between ⊥ and ⊥. Indeed, one has J ⊥ = ⊥. (v) follows from ⊥ = . ✷ E E E E E E Two closed subspaces are said to form a Fredholm pair if both their intersection and the codimension of their sum are ﬁnite dimensional. A frame for a subspace is a partial isometry from some Hilbert space onto it. More details on these Hilbert space concepts are recollected in Appendix B.

Proposition 2 Let and be a Fredholm pair of Lagrangian subspaces with trivial intersection and let Φ and Ψ beE framesF for them respectively. Suppose that = + . Then the oblique projection P with range and kernel is given by H E F E F 1 P = Φ Ψ∗J Φ − Ψ∗ J. Proof. This follows directly from Proposition 41 because JΨ is a frame for ⊥. ✷ b F As already pointed out, ⊥ and do not span in general. If, however, is non-degenerate, then this holds as shown inE the followingE result. K E

Proposition 3 Let be a closed non-degenerate subspace of with frame Φ. Let us suppose E K that 0 σess(Φ∗JΦ). Then 6∈ b = + ⊥ , (1) K E E b and the oblique projection P with range and kernel ⊥ is given by E b E 1 P = Φ(Φ∗J Φ)− Φ∗J. (2)

It satsiﬁes P ∗ = JPJ.

Proof. For (1) one has to show that every φ can be written as φ = Φv + ψ with unique b ∈ K vectors u and ψ ⊥. For that purpose, let us multiply this equation from the left by Φ∗J. As ∈E Φ∗Jψ = 0, this gives Φ∗Jφ = Φ∗J Φ v. Now the non-degeneracy of shows that 0 is not an E

5 eigenvalue of the self-adjoint operator Φ∗JΦ. As there is no essential spectrum by hypothesis, it follows that Φ∗JΦ is invertible. Thus

1 v = Φ∗J Φ − Φ∗J φ . b Then ψ = φ Φ v is by construction in ⊥. As there was no freedom in this construction, uniqueness of −v and ψ is guarenteed. Furthermore,E the projection P is given by Pφ = Φv leading to (2). From this the last formula is now readily deduced. ✷

2.2 The inertia and signature of subspaces

First let us recall that for any self-adjoint operator H = H∗ (identified with a quadratic form), the inertia ν(H)=(ν+, ν , ν0) is defined by the dimensions of the spectral projection of H on − (0, ), ( , 0) and 0 respectively. The inertia is sometimes also called signature, but here ∞ −∞ { } the signature of H is Sig(H)= ν+ ν . Another number used in this context is the Witt index − − min ν+, ν + ν0 (this is equal to the dimension of the maximally isotropic subspaces). If ν0 − vanishes,{ it} is often suppressed in the inertia. If H acts on an infinite dimensional Hilbert space, all entries of the inertia can be infinite, but, of course, one is most interested in cases where some of them are finite. The main result about the inertia used below is Sylvester’s law stating that 1 ν(C∗HC) = ν(H) for any invertible operator C. Furthermore, one has ν(H− ) = ν(H) for any invertible H.

Deﬁnition 3 Let be a subspace of a Krein space ( ,J). If Φ is a frame for , then let us set E K E J = Φ∗JΦ. The inertia ν( )=(ν+( ), ν ( ), ν0( )) and signature Sig( ) of are the inertia E E E − E E E E and signature of J , that is, ν( )= ν(J ) and Sig( )= Sig(J ). E E E E E Sylvester’s law of inertia implies that the deﬁnition of ν( ) and Sig( ) is independent of the choice of the frame Φ. Let us note a few obvious connectionsE betweenE inertia and the other notions introduced above.

Lemma 2 Let be a subspace of . E K (i) isotropic ν+( )= ν ( )=0 E ⇐⇒ E − E (ii) non-degenerate ν ( )=0 E ⇐⇒ 0 E (iii) positive deﬁnite ν0( )= ν ( )=0 E ⇐⇒ E − E (iv) negative deﬁnite ν ( )= ν ( )=0 E ⇐⇒ 0 E + E (v) ν+( )+ ν ( )+ ν0( ) = dim( ) E − E E E The following result provides an alternative way to calculate the inertia and the signature of a non-degenerate subspace.

Proposition 4 Let be a closed non-degenerate subspace of with frame Φ. Let us suppose E K b that 0 σess(Φ∗JΦ). Further let P be the oblique projection given by (2). Then ⊥ is also non-degenerate6∈ and E b b = + ⊥ , ν( ) + ν( ⊥)=(n, n, 0) . K E E E E b 6 Furthermore ν+( ) = ν+(P ∗JP ) , ν ( ) = ν (P ∗JP ) . E − E − b Proof. Let Φ and Ψ be frames for and ⊥. Then (Φ, Ψ) is invertible in and E E K Φ∗JΦ 0 (Φ, Ψ)∗ J (Φ, Ψ) = . 0 Ψ∗JΨ From this the ﬁrst claims can be deduced. As to the last one, let us ﬁrst of all note 1 P ∗JP = JΦ(Φ∗JΦ)− Φ∗J. Therefore 1 ν(P ∗JP ) = ν(Φ(Φ∗JΦ)− Φ∗) .

Because the kernel of Φ∗ has dimension 2n dim( ), it now follows that − E 1 ν(P ∗JP ) = ν((Φ∗JΦ)− ) 0, 0, 2n dim( ) . − − E From this the result follows. ✷

2.3 Lagrangian subspaces This section further analyzes the Grassmanian L( ,J) of J-Lagrangian subspaces. It will be K useful to describe Lagrangian subspaces by Lagrangian frames. Because Lagrangian subspaces are half-dimensional and closed, it is suggestive to choose these Lagrangian frames as linear maps Φ: = with Φ∗Φ= 1 and such that Ran(Φ) is a Lagrangian subspace. This H→K H ⊕ H latter fact is equivalent to Φ∗JΦ=0. The set of Lagrangian frames is a U( )-cover of L( ,J) H K because Φ and Φu span the same subspace for every u U( ). Moreover, ΦΦ∗ is an orthogonal ∈ H projection in with range given by the range of Φ. The complementary orthogonal projection K JΦ(JΦ)∗ projects on the J-Lagrangian subspace with frame JΦ and one has

1 = ΦΦ∗ + JΦ(JΦ)∗ . (3) Clearly, also the set of all orthogonal projections with range given by a J-Lagrangian subspace is in bijection with L( ,J). K Theorem 1 Let Ψ be a fixed Lagrangian frame. For any other Lagrangian frame Φ, introduce bounded operators x and y on by H Φ =Ψ x + J Ψ y , (4) and define the stereographic projection of Φ along Ψ by 1 π (Φ) = (x + y)(x y)− = (x + y)(x y)∗ . (5) Ψ − − Then πΨ is well-defined and unitary, namely πΨ(Φ) U( ). Moreover, one has πΨ(Φ) = πΨ(Φu) for any u U( ) so that π factors to a map on ∈L( ,JH) also denoted by π . It establishes a ∈ H Ψ K Ψ bijection π : L( ,J) U( ) with inverse given by Ψ K → H 1 1 1 π− (u)= Ψ (u + 1) + J Ψ (u 1) , (6) Ψ 2 2 − where the r.h.s.’s of this equation gives one representative in L( ,J). K 7 Proof. Equation (4) can be rewritten as

x Φ = (Ψ,J Ψ) . y Now one directly checks that (Ψ,J Ψ) is unitary. Thus

x = (Ψ,J Ψ)∗ Φ y is a frame, that is x∗x + y∗y = 1. Furthermore, one has

x ∗ x x ∗ 0 1 x 0 = Φ∗JΦ = (Ψ,J Ψ)∗J(Ψ,J Ψ) = = x∗y + y∗x , y y y 1 0 y so that (x y)∗(x y) = 1 . ± ± This shows that πΨ is well-defined and unitary. That the inverse is indeed given by (6) can be readily checked. ✷ 1 1 Remarks A standard choice for a reference J-Lagrangian subspace is Ψ=2− 2 1 . In this case, the stereographic projection is simply denoted by π = π and takes the following particularly Ψ simple form: 1 a π(Φ) = ab− , Φ = . (7) b This also shows that πΨ is indeed a generalization of the standard stereographic projection in R2. If Φ: spans a Lagrangian subspace , but only Φc is a frame for some invertible H→K E c B( ), then one can still define x and y by (4) and the first formula in (5) remains valid, namely ∈ H 1 π( )=(x + y)(x y)− . However, the factors x y are not unitary any more. Furthermore, π E − ± allows to calculate πΨ via α π (Φ) = (√2β)∗π(Ψ)∗π(Φ)(√2β) , Ψ = . (8) Ψ β

As √2β is unitary, this shows that the spectra of πΨ(Φ) and π(Ψ)∗π(Φ) coincide. Let us also note that for any u U( ) ∈ H πΨu(Φ) = u∗πΨ(Φ)u .

In particular, the spectrum of πΨ(Φ) does not depend on Ψ, but only the Lagrangian subspace spanned by it. These spectral properties are of importance in view of the following result. ⋄ The dimension of the intersection of two Lagrangian subspaces can be conveniently read oﬀ from the spectral theory of the associated stereographic projection, as shows the next proposition.

Proposition 5 Let and be two J-Lagrangian subspaces of with Lagrangian frames Φ and Ψ respectively. ThenE F K

dim = dim Ker(Ψ∗J Φ) = dim Ker(π (Φ) 1) = codim( + ) . E∩F Ψ − E F 8 Proof. Let us begin with the inequality of the first equality. Let p N be the dimension of . Then there are two partial isometries≤ v,w : ℓ2( 1,...,p )∈ ∪{∞}such that Φv = Ψw. E∩F { } → H Then Ψ∗JΦw = Ψ∗JΨv = 0 so that the kernel of Ψ∗JΦ is at least of dimension p. Inversely, 2 given an isometry w : ℓ ( 1,...,p ) such that Ψ∗JΦw =0, one deduces that (JΨ)∗Φw =0. As the spans of Ψ and J{Ψ are orthogonal} → H and span all by (3), it follows that the span of Ψw lies in the span of Φ. This shows the other inequality andK hence proves the first equality. Next let us first note that the dimension of the kernel of Ψ∗JΦ does not depend on the choice of the representative. Using the representative given in (6), one finds 1 Ψ∗J Φ = (π (Φ) 1) , (9) 2 Ψ − which implies the second equality. Finally the third equality follows from

codim( + ) = dim ( + )⊥ = dim ⊥ ( )⊥ = dim J J , E F E F E ∩ F E ∩ F and the fact that J is an isomorphism. ✷

Theorem 2 Let and be two J-Lagrangian subspaces with associated Lagrangian frames Φ and Ψ. Then theE followingF are equivalent: (i) and form a Fredholm pair E F (ii) Ψ∗JΦ is a Fredholm operator on H (iii) π (Φ) 1 is a Fredholm operator on Ψ − H (iv) 1 is not in the essential spectrum of πΨ(Φ) The index Ind( , ) associated to the Fredholm pair of Lagrangian subspaces vanishes. E F Proof. The equivalence of (i) and (ii) follows from Theorem 11 in Appendix B. The equivalence of (ii) and (iii) follows from the identity (9). The equivalence of (iii) and (iv) holds for any unitary operator πΨ(Φ), by the same argument showing that a selfadjoint operator is Fredholm if and only if 0 is not in the essential spectrum. The last claim follows immediately from the deﬁnition of the index (see Appendix B) and Proposition 5. ✷

The theorem shows that the spectrum of the unitary πΨ(Φ) allows to determine a distance between the two subspaces. If its eigenvalues are phases close to 0, then one is near an intersection between the subspaces. The link of the angle spectrum σ( , ) between the subspaces (see E F Appendix B for a deﬁnition) to the spectrum of πΨ(Φ) is discussed in the following result.

Proposition 6 Let Φ and Ψ be Lagrangian frames. Then ϕ eıϕ σ(π (Φ)) with ϕ ( π, π] | | σ( , ) . ∈ Ψ ∈ − ⇐⇒ 2 ∈ E F Proof. Using (6), one ﬁnds 1 Ψ∗Φ = (π (Φ) + 1) , 2 Ψ so than 1 Φ∗ΨΨ∗Φ = (π (Φ) + 1)∗(π (Φ) + 1) . 4 Ψ Ψ 9 Now by spectral calculus of π (Φ) the claim follows from 1 eıϕ +1 2 = cos2( ϕ ). ✷ Ψ 4 | | 2 Note that the spectrum of πΨ(Φ) contains more information than the angle spectrum, namely a supplementary sign for each eigenvalue. In other words, a Lagrangian subspace comes with a second Lagrangian frame J orthogonal to , and within the associated quarterF plane splitting, one can deﬁne angles togetherF with an orientation.F

2.4 The Bott-Maslov index in infinite dimensions Given a fixed Lagrangian subspace in ( ,J) and associated frame Ψ for , let us introduce the Fredholm Lagrangian GrassmanianF w.r.t.K by F F FL( , J, ) = L( ,J) ( , ) form Fredholm pair . K F {E ∈ K | E F } FL One is now interested in (continuous) paths γ = (γt)t [t0,t1) in ( , J, ) and in counting the ∈ K F number of points t with non-trivial intersections γt , however, with an orientation as weight. This weighted sum of intersections is then the Bott-Maslov∩F index which will now be defined in detail as a spectral flow. As in the finite dimensional case [Arn], the singular cycle of Lagrangian subspaces with non-trivial intersections is the stratified space defined by

S( ) = S ( ) , S ( ) = FL( , J, ) dim = l . F l F l F E ∈ K F E∩F l 1 [≥

Note that for FL( , J, ) it never happens that dim( )= . Under the stereographic E ∈ K F E∩F ∞ projection πΨ one gets from Theorem 2

π S( ) = Uess( ) , Ψ F H where Uess( ) are those u U( ) satisfying 1 σess(u), and furthermore from Proposition 5 H ∈ H 6∈

π S ( ) = u U( ) 1 σess(u) and dim(ker(u 1)) = l . (10) Ψ l F { ∈ H | 6∈ − } Let now γ = (γt)t [t0,t1) be a (continuous) path as above for which, for sake of simplicity, the ∈ number of intersections t [t , t ) γ S( ) is finite and does not contain the initial point { ∈ 0 1 | t ∈ F } t0. Associated to γ is the path u = π (γ ) Uess( ) , t Ψ t ∈ H for which a spectral flow SF((ut)t [t0,t1)) through 1 is defined in an obvious manner, as recalled ∈ in Appendix E. This spectral flow now defines the Bott-Maslov intersection number or index of the path γ w.r.t. the singular cycle S( ): F

BM(γ, ) = SF (ut)t [t0,t1) . (11) F ∈ Let us collect without detailed proof a few basic properties of the index.

U Proposition 7 For T ( ,J) and γ as above, set Tγ =(Tγt)t [t0,t1). ∈ K ∈ FL (i) Let the path γ +γ′ denote the concatenation with a second path γ′ =(γt)t [t1,t2) in ( , J, ). ∈ K F 10 Then BM(γ + γ′, ) = BM(γ, )+ BM(γ′, ) . F F F (ii) Given a second path γ′ =(γ′) in FL( ′,J ′, ′) where ′ is a Lagrangian subspace in a t [t0,t1) K F F Krein space ( ′,J ′), one has, with denoting the symplectic direct sum, K ⊕

BM(γ γ′, ′) = BM(γ, ) + BM(γ′, ′) . ⊕ F ⊕Fb F F (iii) One has BM(T γ,T )=bBM(γ, b ). F F (iv) For a closed path γ, BM(γ, ) is independent of as long as it remains in FL( , J, ). F F K F 3 Basic analysis of J-unitary operators

Let U( ,J) denote the set of J-unitary operators, namely all invertible T B( ) satisfying K ∈ K T ∗JT = J.

3.1 Möbius action of J-unitaries Proposition 8 U( ,J) is a -invariant group. One has K ∗ a b U( ,J) = a∗a c∗c = 1 , d∗d b∗b = 1 , a∗b = c∗d (12) K c d − − a b = aa∗ bb∗ = 1 , dd∗ cc∗ = 1 , ac∗ = bd∗ , c d − −

1 1 and in this representation a and d are invertible and satisfy a− 1, d− 1. Also 1 1 1 1 k k ≤ k k ≤ a− b < 1, d− c < 1, bd− < 1 and ca− < 1. k k k k k k k k 1 1 Proof. The group property is obvious. Inverting T ∗JT = T shows T − J(T ∗)− = J so that J = T JT ∗. The relations in (12) are equivalent to T ∗JT = T and T JT ∗ = T . The fact that 1 1 a is invertible follows from aa∗ 1. Furthermore aa∗ bb∗ = 1 implies that a− b(a− b)∗ = 1 1 1 ≥ − 1 a− (a− )∗ < 1, so that a− b < 1. The same argument leads to the other inequalities. ✷ − k k The following result is well-known [KS].

Theorem 3 The group U( ,J) acts on the Siegel disc D( ) = u B( ) u < 1 and also K H { ∈ H |k k } on U( ) by Möbius transformation denoted by a dot and deﬁned by: H

a b 1 u = (au + b)(cu + d)− , u B( ) . c d · ∈ H Proof. One ﬁrst has to show that for u B( ) with u < 1 and T U( ,J) the inverse in ∈ H k k ∈ K 1 the Möbius transformation T u is well-deﬁned. By Proposition 8, (cu + d) = d(1 + d− cu) is · indeed invertible. Then the identities of Proposition 8 imply

(cu + d)∗ (cu + d) (au + b)∗ (au + b) = 1 u∗ u . (13) − − 11 1 1 Now multiplying (13) from the left by ((cu + d)∗)− and the right by (cu + d)− and using 1 u∗u> 0 shows (T u)∗(T u) < 1 so that T u D( ). By the same argument, if u U( ), − · · · ∈ H ∈ H then T u U( ). A short algebraic calculation also shows that (T T ′) u = T (T ′ u). ✷ · ∈ H · · · The action of U( ,J) on U( ) is the stereographic projection of the natural geometric action of U( ,J) on the LagrangianK H Grassmannian L( ,J). This natural action sends a subspace K K L( ,J) to T L( ,J). Let Φ be a frame for , then T Φ is not a again a frame, but E ∈ K E ∈ K1 E 2 T Φ = T Φ (T Φ)∗(T Φ) − is a frame for T . Thus one indeed has using the above Möbius action:· E π T Φ = T π Φ . (14) · · The following calculation will turn out to be useful later on.

at bt Proposition 9 Let t Tt = be a differentiable path in U( ,J) and u U( ). Then 7→ ct dt K ∈ H 1 u ∗ u 1 (T u)∗∂ (T u) = (c u + d )− ∗ T ∗J∂ T (c u + d )− . t · t t · t t 1 t t t 1 t t 1 Proof. For sake of notational simplicity, let us suppress the index t. Using (T u)∗ =(T u)− · · and the laws of operator differentiation, one finds

1 1 (T u)∗∂ (T u) = (T u)∗ ∂(au + b) (cu + d)− ∂(cu + d) (cu + d)− · t · t · − 1 1 = (cu + d)− ∗ (au + b)∗∂(au + b) (cu + d)∗∂(cu + d) (cu + d)− . − This directly leads to the identity. ✷

3.2 Basic spectral properties and Riesz projections of J-unitaries Most results in the section go at least back to [Lan], and can also be found in [Bog]. Standard notations for the spectrum as well as the continuous, discrete, point and residual spectrum are used.

Proposition 10 Let T be a J-unitary. 1 (i) σ(T )= σ(T )− 1 − (ii) σc(T )= σc(T ) 1 − (iii) σd(T )= σd(T ) 1 (iv) z σr(T ) implies z− σp(T ) ∈ 1 ∈ (v) z σ (T ) implies z− σ (T ) σ (T ) ∈ p ∈ p ∪ r (vi) σ (T ) S1 = r ∩ ∅ Proof. The main identity used in the proof is

1 1 1 T z1 = J ∗((T ∗)− z1)J = zJ(T ∗)− (T z− 1)∗J. − − −

12 (i) As J and T ∗ are invertible (with bounded inverses), invertibility of T z1 is equivalent to 1 − invertibility of T z− 1. (ii) Recall that the continuous spectrum consists of those points z for which T z1 is bijective− and has dense range. Again these properties are conserved in the above − 1 identity. (iii) Due to (i) only remains to show that the isolated points z, z− σ(T ) both have ﬁnite same multiplicity. But this follows from Proposition 11 proved independently∈ below, when combined with Proposition 42(v). (iv) Let z σp(T ) with eigenvector φ. Then by (i) one has 1 1 ∈ 1 z− σ(T ) and wants to exclude z− σc(T ), namely that T z− 1 is bijective with dense range.∈ Let us suppose the contrary. Then∈ there exists a ψ with− ∈K 1 1 1 1 1 0 = φ∗J(T z− 1)ψ = φ∗((T − )∗ z− 1)Jψ = z− ((z1 T )φ)∗(T − )∗Jψ . 6 − − − But this is a contradiction to Tφ = zφ. (v) Let z σ (T ). Then there exists some Jψ such ∈ r ∈K that for all φ ∈K 1 1 0 =(Jψ)∗(T z1)φ = z ((T z− 1)ψ)∗(T ∗)− Jφ . − − 1 But this shows that T ψ = z− ψ. (vi) is a corollary of (iv) and (v). ✷ Let T be a J-unitary and ∆ σ(T ) be a separated spectral subset, namely a closed subset ⊂ which has trivial intersection with the closure of σ(T ) ∆. Then P∆ denotes the Riesz projection of T on ∆ and furthermore = Ran(P ) and =\ Ker(P ). If ∆ = λ , let us also simply E∆ ∆ F∆ ∆ { } write Pλ, λ and λ. The deﬁnition of P∆ and a few basic properties which do not pend on the KreinE space structureF are recalled in Appendix C. Next follows a list of properties of Riesz projections of J-unitaries. Proposition 11 Let T be a J-unitary and ∆ a separated spectral subset. Then

(P )∗ = J ∗ P −1 J, ⊥ = J −1 . ∆ (∆) F∆ E(∆) 1 Proof. First of all, let us note that indeed (∆)− is in the spectrum of T by Proposition 10, and 1 thus by the spectral mapping theorem one also knows that ∆ is in the spectrum of T − . Let us take the adjoint of the deﬁnition (53) of the Riesz projection:

dz 1 (P )∗ = (z T ∗)− , ∆ 2πı − IΓ where Γ is the complex conjugate of Γ, hence encircling ∆ instead of ∆. It is also positively oriented even though the complex conjugated of the path Γ would have inverse orientation, but the imaginary factor compensates this. Thus P∆(T )∗ = P∆(T ∗) if one adds the initial operator 1 as an argument to the Riesz projection. Next let us use T ∗ = J ∗T − J:

dz 1 1 (P )∗ = J ∗ (z T − )− J. ∆ 2πı − IΓ Now Proposition 42(ii) concludes the proof of the ﬁrst identity. As to the second,

⊥ = Ker(P )⊥ = Ran(P ∗ ) = Ran(JP −1 J) = J −1 , F∆ ∆ ∆ (∆) E(∆) so that the proof is complete. ✷ When combined with Proposition 42(v) one obtains the following.

13 1 Corollary 1 Let ∆ be a disjoint separated spectral subset of a J-unitary satisfying ∆ ∆− = . ∩ ∅ Then dim( ) = dim( −1 ). E∆ E∆

Proposition 12 Let ∆ and ∆′ be disjoint separated spectral subsets of a J-unitary satisfying 1 ∆ ∆− = . Then and ′ are J-orthogonal. Similarly, ⊥ and ⊥′ are J-orthogonal. ′ ∩ ∅ E∆ E∆ F∆ F∆ Proof. By Propositions 11 and 42, one has

′ −1 ′ P∆∗ JP∆ = JP∆ P∆ = 0 .

But this is implies that and ′ are J-orthogonal. The other claim is proved similarly. ✷ E∆ E∆ Proposition 13 Let T be a J-unitary and suppose that its spectrum is decomposed σ(T ) = L 1 l=1 ∆l into disjoint separated spectral subsets ∆l satisfying ∆l = (∆l)− . Then, with pairwise J-orthogonal subspaces, S = + ... + . (15) K E∆1 E∆L Moreover, each summand is non-degenerate. b b

Proof. The fact that = ∆1 + ... + ∆L follows from Proposition 42(iv). Moreover, Propo- sition 12 implies that theK subspacesE areE pairwise J-orthogonal. To prove the last claim, let us suppose that φ ∆ is in the kernel of J . Then φ is J-orthogonal to ∆ and hence by the l E∆l l above J-orthogonal∈E to all . But as J is non-degenerate,| this implies thatEφ =0. ✷ K

3.3 Signatures of generalized eigenspaces of normal eigenvalues In this section, let us fix T U( ,J) which has K normal eigenvalue pairs off the unit circle and k normal eigenvalues on∈ the unitK circle. The eigenvalues off the circle come in pairs and are 1 denoted by (λ , (λ )− ) with λ < 1 and l =1,...,K, and further let the eigenvalues on the circle l l | l| be λ′ ,...,λ′ . Let us denote the span of all associated generalized eigenspaces by , a notation 1 k E= which then coincides with the choice made for essentially S1-gapped operators in Section 5. Now Proposition 13 can be applied and it yields

′ ′ = = λ + ... + λ + λ + −1 + ... + λ + −1 , (16) E E 1 E k E 1 E(λ1) E K E(λK ) with pairwise J-orthogonalb non-degenerateb b summands whichb b all have trivial intersection. As to the dimensions, one has

k K

dim( =) = dim( λ′ )+2 dim( λ ) . E E l E l l l X=1 X=1 Let us set n = dim( ). Let Φ be a frame for and introduce the n n matrices = E= = E= = × = J= = Φ=∗ JΦ= and T= = Φ=∗ T Φ=. Then T= is J=-unitary. For sake of concreteness it can be useful to choose an adequate basis change N corresponding to (16) which brings J= into a particularly simple normal form. It is important to note that N is in general not unitary.

14 1 Proposition 14 There exists an invertible n n matrix N such that both N − T N and = × = = N ∗J=N are block diagonal with blocks of the size of the summands in (16). Moreover, N can be chosen such that the first blocks of N ∗J=N corresponding to eigenvalues on the unit circle are diagonal with diagonal entries equal to 1 or 1 and that the blocks of N ∗J=N corresponding to 0 ı − eigenvalue pairs off the unit circle are ı −0 . Proof. Let Ψ=(ψ1,...,ψn= ) be composed by basis vectors of the summands in (16), respecting 1 the order of (16). Then define N ′ by Φ=N ′ = Ψ. Then clearly (N ′)− T=N ′ is block-diagonal, but because these summands are J-orthogonal, it follows that also (N ′)∗J=N ′ is block-diagonal. Moreover, (N ′)∗J=N ′ is self-adjoint and has no vanishing eigenvalue. In a second step, one can diagonalize the blocks corresponding to eigenvalues on the unit circle and then multiply with adequate invertible diagonal matrices in order to produce eigenvalues 1 and 1. Finally, by − Proposition 12 (applied to ∆ = ∆′ = λl ), the blocks corresponding to eigenvalues off the unit circle are of the form { } 0 A , A∗ 0 where, moreover, A is invertible by Proposition 13, as otherwise the block would be degenerate. Now the basis change

1 0 ∗ 0 A 1 0 0 ı 1 1 1 = − 0 ı A− A∗ 0 0 ı A− ı 1 0 − − can be done within this block. Combining these blockwise operations one obtains N from N ′. ✷

Definition 4 Let λ be a normal eigenvalue of T U( ,J) with generalized eigenspace . Its ∈ K Eλ inertia (ν+(λ), ν (λ)) is defined as follows: − (i) If λ =1, then the inertia is the inertia of λ, namely that of the form J on λ. | | E Eλ E (ii) If λ > 1, then the inertia is equal to (dim( ), 0). | | Eλ (iii) If λ < 1, then the inertia is equal to (0, dim( )). | | Eλ Eigenvalues for which ν+(λ)=0 or ν (λ)=0 are called negative and positive definite respectively. An eigenvalue which is not definite is− called indefinite or of mixed inertia.

In the russian literature [Kre, GL, YS], the terms of first kind, of second kind and of mixed kind are used instead of positive definitive, negative definite and indefinite. Just as for quadratic forms, the inertia is also sometimes called signature and in the present context Krein signature. The inertia of the eigenvalues on the unit circle can be obtained by counting the entries 1 and 1 of the corresponding block in Proposition 14 and there is indeed no vanishing eigenvalue (which− is where ν0(λ)=0 is suppressed in the notation). For eigenvalues off the unit circle the definition of the inertia is rather a convention, which is, however, consistent with Proposition 14 because the inertia of blocks corresponding to + −1 with λ =1 is (dim( ), dim( )). Eλ Eλ | | 6 Eλ Eλ The following result shows that for eigenvalues on the unit circle the definiteness can be checked by only looking at eigenvectors (and hence not the generalized eigenvectors, often also called root vectors).

15 Proposition 15 Let λ be a unit eigenvalue of T U( ,J). Then ∈ K ν (λ) =0 φ∗Jφ > 0 for all eigenvectors φ of λ . ± ⇐⇒ ∓ Proof. The implication “= ” is clear. For the converse, let us show that the condition on the ⇒ r.h.s. actually implies that λ only consists of eigenvectors so that again the definiteness follows. Hence let us suppose thatE there is a non-trivial Jordan block, namely that there are vectors φ, ψ such that Tφ = λφ and T ψ = λψ + φ. Then ∈Eλ 2 φ∗Jψ = φ∗T ∗JT ψ = (λφ)∗J(λψ + φ) = λ φ∗Jψ + λφ∗Jφ . | | Hence 2 λφ∗Jφ = (1 λ ) φ∗Jψ = 0 . −| | But this shows φ∗Jφ =0, which is a contradiction to the hypothesis. ✷ The previous proof actually also shows the following result. Corollary 2 Let λ be a unit eigenvalue of T U( ,J). ∈ K (i) If there is a non-diagonal Jordan block for λ, then λ is indefinite and there exists an eigenvector φ such that φ∗Jφ =0. (ii) If λ is definite, then all Jordan blocks are diagonal. The following is a conservation law of the inertia of all eigenvalues in . E= Proposition 16 The inertia of T U( ,J) as above satisfy ∈ K ν (λ) = ν ( =) . (17) ± ± E λ σ(T=) ∈X Proof. The inertia of the quadratic form J= is (ν+( =), ν ( =)). By Sylvester’s law it is − also equal to the sum of the inertia of the summands itE in theE decomposition (16). Therefore Definition 4 completes the proof. ✷ Proposition 17 The inertia ν(λ) of normal eigenvalues of T U( ,J) depends continuously on the T . ∈ K Proof. First of all, let us recall [Kat, IV.3.5] that the normal eigenvalues of T depend continuously on T . Hence the result can be restated as follows. For any continuous path t R T (t) with T (0) = T , the eigenvalues λ(t) can be labelled (at eigenvalue crossings) such that∈ the7→ inertia ν(λ(t)) is constant in the sense that the sum of all inertia of colliding eigenvalues is constant. Clearly it is sufficient to prove this continuity at one point, say t = 0. As the eigenvalues of T off the unit circle stay off the unit circle for t sufficiently small, their inertia are preserved as well. Therefore one only has to consider eigenvalues of T on the unit circle. Hence let λ be such an eigenvalue and P (t) be the finite dimensional Riesz projection of T (t) on all eigenvalues of T (t) converging to λ as t 0. By [GK, Theorem 3.1], the dimension of Ran(P (t)) is constant → in t. Let Φ(t) be a frame for P (t). Then the quadratic form Φ(0)∗JΦ(t) has precisely ν+(λ) positive eigenvalues and ν (λ) negative eigenvalues, and 0 is not an eigenvalue because λ is − E non-degenerate. By continuity of the eigenvalues of finite dimensional matrices, these properties remain conserved for small t. But ν(Φ(t)∗JΦ(t)) is precisely the sum of the inertias of all eigenvalues converging to λ as t 0. ✷ → 16 3.4 Krein stability analysis Proposition 18 Let λ be a simple normal unit eigenvalue of T U( ,J). Then there is a neighborhood of T such that the eigenvalue close to λ stays on the unit∈ circle.K Proof. This follows from the symmetry of the spectrum of T stating that, if λ is an eigenvalue, 1 so is λ− . Thus if λ would leave the unit circle, it would have to lead to two eigenvalues. ✷ Proposition 18 is of elementary nature. However, it does not allow to say anything about structural stability in presence of degenerate eigenvalues appearing. These issues were first addressed by Krein, and the following result is basically due to him [Kre] and Gelfand and Lidskii [GL]. Theorem 4 Let λ be an normal unit eigenvalue of T U( ,J). Suppose that λ is definite. Then there is a neighborhood of T in which all operators∈ haveK the property that all eigenvalues close to λ lay on the unit circle and have diagonal Jordan blocks. First proof. Let us suppose the contrary. Then there exists a sequence of J-unitary operators 1 (Tn)n 1 converging to T with corresponding eigenvalues (λn)n 1 converging to λ S such that ≥ ≥ ∈ for each n either (i) λ = 1 or (ii) λ = 1 with a non-diagonal Jordan block. Let φ be | n| 6 | n| n ∈ K corresponding normalized eigenvectors, namely Tnφn = λnφn, such that φn∗ Jφn =0 for all n 1. Indeed, this holds in case (i) for all eigenvectors due to Proposition 12 (applied with two equal≥ eigenvalues) and in case (ii) it can be assured due to Corollary 2(i). By weak compactness of the set of unit vectors, there now exists a subsequence also denoted by φn that converges weakly to a vector φ. Then, for any ψ , ∈K ψ∗Tφ = lim ψ∗Tφn = lim ψ∗(T Tn)φn + ψ∗Tnφn = λ ψ∗φ . n n →∞ →∞ − Thus Tφ = λφ and similarly φ∗Jφ =0 so that λ would not be definite. ✷ Second proof. Let t T (t) be the path of J-unitaries with T (0) = T . By Proposition 17 the inertia has to stay definite7→ for t sufficiently small. Hence it cannot leave the unit circle because this would lead to an indefinite inertia of the eigenvalue group by Proposition 14. ✷ Let us point out that both non-trivial Jordan blocks with eigenvalues on S1 and diagonal Jordan blocks with mixed inertia are unstable in the sense that in any neighborhood of them are operators with eigenvalues off the unit circle. This can be shown by adapting the arguments in finite dimension from [YS].

3.5 The twisted graph of a J-unitary This section is an improved and generalized version of what is given in [SB2]. Let us associate to an operator T on = a new operator T = 1 T on = where the denotes the symplectic checkerK boardH ⊕ sum H given by ⊕ K K⊕K ⊕ b b ab 0 b 0 b a b a′ b′ 0 a′ 0 b′ =   . (18) c d ⊕ c′ d′ c 0 d 0  0 c′ 0 d′  b     17 If T U( ,J), then T = 1 T is in the group U( , J) of operators conserving the form J = ∈ K ⊕ K J J. The stereographic projection from L( , J) to U( ) defined as in (7) is denoted by π. ⊕ K K Furthermore, one can alsob defineb the symplectic sumb ofb frames by b b b b a 0 b a a′ 0 a′ =   . a, b, a′, b′ B( ) . (19) b ⊕ b′ b 0 ∈ H  0 b′  b     If Φ and Ψ are J-Lagrangian, then Φ Ψ is J-Lagrangian. Let us use a particular frame which is not of this form: ⊕ 1 b b 0 1 1 0 Ψ0 =   . (20) √2 1 0  0 1  b     One has Ψ0∗ J Ψ0 =0 so that Ψ0 defines a J-Lagrangian subspace. Moreover, 1 0 1 π(Ψ0) b b b π(Ψ0)b = ,b Ψ0 = . (21) 1 0 √2 1 b b Note that in our notationsb b the hat always designatesb objects in the doubled Krein space , but, moreover, T is a particular operator in U( , J) associated to a given T U( ,J). Now asK Ψ is K ∈ K 0 J-Lagrangian, so is T Ψ0 and one can therefore associate a unitary π(T Ψ0) to it. This unitaryb will turn outb to be particularly useful forb theb spectral analysis of T U( ,J) (see Theoremb 6 ∈ K below).b b b b b b Theorem 5 To a given T U( ,J) let us associate a unitary V (T ) by ∈ K

V (T ) = π(Ψ )∗ π(T Ψ ) U( ) . (22) 0 0 ∈ K If T = a b , then c d b b b b b 1 1 1 1 a bd− c bd− (a∗)− bd− V (T ) = − 1 1 = 1 1 . d− c d− d− c d− − − Furthermore V (T ) = πb (T Ψ ). The map T U( ,J) V (T ) U( ) is a continuous dense Ψ0 0 embedding with image ∈ K 7→ ∈ K b b b α β U( ) α, δ invertible . (23) γ δ ∈ K ∈

Proof. By (12) one has d∗d 1 so that d is invertible (similarly a is invertible). Now by (14), ≥ 1 0 0 0 0 a 0 b 0 1 T π(Ψ0) =   , · 0 0 1 0 · 1 0  0 c 0 d  b     18 so that writing out the Möbius transformation, one gets

1 1 0 1 1 0 − 0 1 d− 0 d 0 T π(Ψ0) = = 1 . · a b c d a b 0 d− c 1 − From thisb the ﬁrst formula for V (T ) follows, and the second results from the relations in U( ,J). Clearly its upper left and lower right entries, the matrices denoted by α and δ in (23),K are invertible. It can readily be checked that V (T ) is equal to πb (T Ψ ) and not just unitarily Ψ0 0 equivalent to it as in (8). Finally let us show that the map T U( ,J) V (T ) b U(b )b is surjective onto the set ∈ K 7→ ∈ K1 1 1 (23). Indeed, given an element of this set, it is natural to set d = δ− , b = βδ− , c = δ− γ and 1 − a = α βδ− γ. With some care one then checks that the equations in (12) indeed hold. ✷ − Proposition 19 Given T U( ,J), one has ∈ K 1 1 V (T )− = V (T )∗ = V (T − ) = JV (T ∗) J.

1 Proof. Using T − = JT ∗J, one ﬁnds

1 1 1 1 a− c∗(d∗)− a− a− b V (T ∗) = 1 1 = 1 1 , (d∗)− b∗ (d∗)− ca− (d∗)− − − and 1 1 1 1 1 a− a− b a− c∗(d∗)− V (T − ) = 1 − 1 = 1 − 1 . (24) ca− (d∗)− (d∗)− b∗ (d∗)− This shows the claim. ✷

The following result justiﬁes the above construction and, in particular, the choice of Ψ0.

Theorem 6 Let T and V (T ) be as in Theorem 5. Then b geometric multiplicity of 1 as eigenvalue of T = multiplicity of 1 as eigenvalue of V (T ) , and the eigenvectors coincide.

Proof. One then has for all vectors v, v′,w,w′ ∈ H w v w v V (T ) = ′ T = ′ , w v ⇐⇒ v w ′ ′ as can readily be seen by writing everything out:

1 1 a bd− c bd− w v′ a b w v′ − 1 1 = = . d− c d− w v ⇐⇒ c d v w − ′ ′ In particular, studying an eigenvalue λ of T one has w λw w w V (T ) = T = λ , (25) λ v v ⇐⇒ v v 19 or similarly eigenvalues λ of V : w w w λw V (T ) = λ T = , (26) w w ⇐⇒ λw w ′ ′ ′ ′ λw which means that T conserses the length of w′ . Both equations are particularly interesting in the case λ =1: w w w w V (T ) = T = . (27) v v ⇐⇒ v v This equivalence proves the theorem. ✷ Remark An alternative proof of Theorem 6 is obtained by studying the Bott-Maslov intersection of T Ψ0 with Ψ0. Suppose that these frames have a non-trivial intersection. This means that v v′ there are vectors v,w,v′,w′ such that such Ψ0 w = T Ψ0 w′ . The first and third line of ∈ H v v thisb vectorb equalityb imply w = w′ and v = v′, the other two that T = . This shows w w b b b geometric multiplicity of 1 as eigenvalue of T = dim T Ψ Ψ . 0 K ∩ 0 K But now Proposition 5 can be applied to calculate the r.h.s., and thisb b leads againb to a proof of Theorem 6. ⋄ Remark If w is an eigenvector of T U( ,J) with eigenvalue off the unit circle, then w = v ∈ K k k v . Indeed, by (25) and the fact that V (T ) is unitary and therefore isometric follows that k k w 2 + λ 2 v 2 = λ 2 w 2 + v 2 . k k | | k k | | k k k k As λ =1 the claim follows. This fact is compatible with the fact that the eigenspace with λ | | 6 Eλ a normal eigenvalue off the unit circle is J-orthogonal to itself (see Proposition 12). However, in the above argument it is not required that λ is normal. ⋄ Theorem 6 as well as the connection between eigenvectors can easily be adapted to study other eigenvalues on the unit circle. Indeed, if T v = zv for z S1, then also (z T )v = v. But the operator z T is also J-unitary so that one can apply the above∈ again to construct an associated unitary. This shows the following.

Proposition 20 Let T = a b be a J-unitary and set, for z S1, c d ∈ 1 1 z (a∗)− bd− V (z T ) = π(Ψ0)∗ π(z T Ψ0) = 1 1 . (28) d− c zd− − Then b b b c b geometric multiplicity of z as eigenvalue of T = multiplicity of 1 as eigenvalue of V (z T ) .

Therefore, the unitaries V (z T ) are a tool to study eigenvalues of T which lie on the unit circle. Let us focus again on z = 1. Theorem 6 concerns the kernel of V (T ) 1. It is natural − to analyze how much more spectrum V (T ) has close to 1, or, what is equivalent, how much 1 spectrum the self-adjoint operator e(V (T )) = 2 (V (T )+V (T )∗) has close to 1. For this purpose let us calculate e(V (T )). ℜ ℜ 20 Proposition 21 Let T be a J-unitary and V (T ) as above. Then

1 e(V (T )) = (1 + T ) 1 + T ∗T − (1 + T )∗ 1 . (29) ℜ − Proof. Let us begin by calculating 1 1 e(V (T )) = (V (T )+ V (T )∗) = (V (T )+ 1)(V (T )+ 1)∗ 1 . ℜ 2 2 − 1 Now V (T )= π(Ψ0)∗ π(T Ψ0). The frame Ψ0 deﬁned as in (20) satisﬁes Ψ0∗Ψ0 = . On the other hand, T Ψ0 is not a frame, but b b b b b b b b 1 2 b b Φ = T Ψ0 (T Ψ0)∗T Ψ0 − . is a frame with range equal to theb rangeb ofb T Ψb0.b Furthermore,b b using the representation formula (6) for Lagrangian frames by their unitaries as well as (21), one shows that there is a unitary U U( ) such that b b ∈ K ∗ 1 π(T Ψ0) π(Ψ0) 1 U Φ∗Ψ = = (V (T )∗ + 1) 0 2 1 1 2 b b b b b Replacing this in the aboveb b shows that

1 e(V (T )) = 2 Ψ∗Φ Φ∗Ψ 1 = 2 Ψ∗T Ψ (T Ψ )∗T Ψ − (T Ψ )∗Ψ 1 . ℜ 0 0 − 0 0 0 0 0 0 − Hence remains to calculateb b theb b appearing products:b b b b b b b b b b

0 1 ∗ 0 1 1 a b a b 1 1 (T Ψ0)∗T Ψ0 =     = ( + T ∗T ) , 2 1 0 1 0 2 c d c d b b b b         and 0 1 ∗ 0 1 1 1 0 a b 1 1 (Ψ0)∗T Ψ0 =     = ( + T ) . 2 1 0 1 0 2 0 1 c d b b b     Replacing shows the claim.     ✷

at bt Proposition 22 Let t Tt = be a diﬀerentiable path in U( ,J). Then 7→ ct dt K 1 0 ∗ 1 0 V (Tt)∗∂tV (Tt) = 1 1 Tt∗J∂tTt 1 1 . d− c d− d− c d− − t t t − t t t

21 Proof. First of all,

V (T )∗∂ V (T ) = π(T Ψ )∗ ∂ π(T Ψ ) = T π(Ψ )∗ ∂ T π(Ψ ) . t t t t 0 t t 0 t · 0 t t · 0 Now one can apply Proposition 22 inb b the Kreinb spaceb ( ,bJ): b b b b b K b b

V (Tt)∗∂tV (Tt) = A∗ Ψ0∗Tt∗Jb ∂tbTtΨ0 A, where the second identity in (21) was also used andb Ab isb givenb b by

1 0 0 1 0 − 1 0 A = π(Ψ0)+ = 1 1 . 0 c 0 d d− c d− t t − t t t Now the result follows from consecutivelyb b using the identities

T ∗J ∂ T = 0 T ∗J ∂ T , Ψ∗ 0 T Ψ = T, t t t ⊕ t t t 0 ⊕ 0 ✷ in the above. b b b b b b b As a further preparation for applications below, let us calculate the derivative of V (z T ) w.r.t. the phase z = eıt S1 and thus set ∈

1 ıt ıt Q (T ) = V (e− T )∗ ∂ V (e− T ) . (30) z ı t This is a self-adjoint operator on and can hence also be interpreted as a quadratic form which by the following result is non-degenerate.K

a b Proposition 23 Let T = c d be a J-unitary. Then

1 1 (a∗)− 0 ∗ 1 z b (a∗)− 0 Qz(T ) = 1 − − 1 0 d− z b∗ 1 0 d− − 1 1 1 (a∗a)− za− bd− = − 1 1 − 1 . (za− bd− )∗ (dd∗)− − Moreover, 0 σ(Q (T )). 6∈ z Proof. Deriving (28) and using (24) for z T shows

1 1 1 z a− a− b z (a∗)− 0 Qz(T ) = 1 − 1 − 1 , (d∗)− b∗ z (d∗)− 0 zd− from what the identity can readily be deduced. Furthermore, for the block operator in the middle v of the expression for Q (T ), a vector in the kernel has to verify v +zbw =0 and zb∗v w =0, z w − so that (1 + bb∗)v =0 and hence v = w =0. Using Weyl sequences one checks that indeed there can be no spectrum at all close to 0, a property that is stable under conjugation with an invertible operator. ✷

22 4 S1-Fredholm operators

4.1 Deﬁnition and basic properties The deﬁnition and basic properties of Fredholm operators are recalled in Appendix A.

Deﬁnition 5 Let T U( ,J) and z S1. Then T is called z-Fredholm if and only if T z 1 is a Fredholm operator,∈ andK T is called∈S1-Fredholm if and only if T z 1 is a Fredholm operator− for all z S1. The set of z-Fredholm and S1-Fredholm operators are− denoted by FU( , J, z) and FU( ,J)∈respectively. K K Let us collect a few ﬁrst facts about such operators.

Proposition 24 Let z S1 and T FU( , J, z). Then the following holds. ∈ ∈ K (i) If S U( ,J) is such that S T is compact, then S FU( , J, z). ∈ K − ∈ K (ii) One has Ind(T z 1)=0. − (iii) There exists an r > 0 such that T ζ 1 is Fredholm for all ζ C with ζ z

(T z 1)(T z 1)∗ = (TJ)(T z 1)∗(T z 1)(TJ)∗ , (31) − − − − implies that also (T z 1)∗ is essentially bounded from below. Therefore again Proposition 39 − concludes the proof. (iv) The identity (31) for S instead of T implies that (S z 1)∗ is essentially bounded from below if and only if S z 1 is essentially bounded from below. Therefore− Corollary 4 − concludes the proof. (v) This follows from the fact that the property of being essentially bounded from below, see (iv), is stable under small perturbations. ✷

Proposition 25 Let T FU( ,J). Then the following holds. ∈ K (i) If S U( ,J) is such that S T is compact, then S FU( ,J). ∈ K − ∈ K (ii) One has Ind(T z 1)=0 for all z S1. − ∈ h h (iii) There exists an h> 0 such that T ζ 1 is Fredholm for all z C with e− < z < e . − ∈ | | (iv) S U( ,J) is z-Fredholm if and only if S z 1 is essentially bounded from below z S1. ∈ K − ∀ ∈ (v) FU( ,J) is open in U( ,J). K K 1 (vi) T FU( ,J) is equivalent to T ∗ FU( ,J) as well as T − FU( ,J). ∈ K ∈ K ∈ K (vii) For T FU( ,J), σ(T ) S1 σ (T ) and each eigenvalue has ﬁnite multiplicity. ∈ K ∩ ⊂ p

23 Proof. (i)-(v) are obvious from the above. (vi) This follows immediately from the identities 1 1 T ∗ z1 = (T z1)∗ and T − z1 = T − z(T z) and the facts that the adjoint of a Fredholm− operator− is Fredholm,− and the− product of− a Fredholm operator with an invertible operator is Fredholm. (vii) Combining item (ii) with Proposition 40 shows that there is neither continuous nor residual spectrum on S1. (That there is no residual spectrum on S1 also follows from Proposition 10(vi).) The second claim is part of the deﬁnition of T z 1 being a Fredholm − operator. ✷ Let us point out that the product ST of S, T FU( ,J) is in general not S1-Fredholm. 1 ∈ K Indeed, the product T T − = 1 is a counterexample. It is reasonable to expect that there is not too much spectrum on S1 for a S1-Fredholm operator. In fact, this is not true as shows the example in Section 6.4. On the other hand, there is always little spectrum of V (z T ) near 1 as shown in the following result. Theorem 7 Let T U( ,J). Then T is z-Fredholm if and only if 1 is not in the essential ∈ K spectrum of V (z T ), or equivalently σess( eV (z T )) < 1. Therefore, ℜ

T FU( ,J) max σess( eV (z T )) < 1 . ∈ K ⇐⇒ z S1 ℜ ∈ Proof. The fact that 1 is not in the essential spectrum of V (z T ) is equivalent to V (z T ) 1 = − π(Ψ )∗π(z T Ψ ) 1 being a Fredholm operator. By Theorem 2 this is in turn equivalent to 0 0 − the fact that the subspaces Ran(Ψ0) and Ran(z T Ψ0) form a Fredholm pair of subspaces in 4 4 b =b b⊗ .c Reorderingb the ﬁrst three components 1, 2, 3 in ⊗ to 3, 1, 2, one obtains equivalence K ∼ H 1 1 H 4 with Ran( 1 ) and Ran( z T ) beingb a Fredholmc pairb of subspaces in ⊗ . As a frame of the 1 1 1 1H 1 1 orthogonalb complement of Ran( ) is and a frame for Ran( ) is (1 + T T ) 2 , 1 √2 1 z T z T ∗ − − 1 1 1 2 Theorem 11 shows equivalence with ( +T ∗T )−2 ( zT ∗) being a Fredholm operator on ⊗ = . As multiplication by an invertible operator does not− change the Fredholm property, oneH concludesK that 1 σess(V (z T )) is equivalent to T z1 being a Fredholm operator. ✷ 6∈ − The gap in the essential spectrum of V (T ) is studied in more detail in Section 4.3.

4.2 Intersection index for paths of z-Fredholm operators FU U Let Γ=(Tt)t [t0,t1] be a path in ( , J, z). Then V (z Tt)t [t0,t1] is a path in the set ess( ) of ∈ K ∈ K unitaries U U( ) with 1 σess(U). Under the same transversality conditions as in Section 2.4 and Appendix∈ EK it is then possible6∈ to deﬁne an intersection index IN(Γ, z) by using the spectral ﬂow of t [t , t ] V (z T ). In other words, ∈ 0 1 7→ t

IN(Γ, z) = SF (V (z Tt)t [t0,t1] = BM z Γ Ψ0, Ψ0 , ∈ FL where z Γ Ψ0 =(z Tt Ψ0)t [t0,t1] is (after normalization) a path in the ( , J, Ψ0). The intersec- ∈ b b K tion index IN(Γ, z) counts the number of solutions Ttφt = zφt along the path, but weighted by an orientationb givend by theb inertia. If nothing is known about these inertia, theb b intersectionb index is really of little use. But in concrete situations such information may be available, for example by using Proposition 22. Examples are the solutions of a Hamiltonian system (Section 6.1), concrete paths (Sections 6.2, 6.5 and 6.8) as well as the more structural result given in Section 5.3.

24 4.3 The essential gap of V (T ) The aim of the following results is to make the gap in the spectrum and essential spectrum of V (T ) more quantitative and to provide tools to check whether a given J-unitary is S1-Fredholm. This section is not crucial for the understanding of the sequel. For an arbitrary operator A on some Hilbert space, let us set

1 1 2 2 g(A) = min σ 1 + A∗A − (1 A)∗(1 A) 1 + A∗A − (32) − − = sup g 0 g 1 + A∗A < (A 1)∗(A 1) , (33) ≥ − − and similarly for gess(A) where in (33) one asks the operator inequality to hold only on a subspace of ﬁnite codimension. This quantity satisﬁes 1 1 1 g(T ) , 1+ T 2 1 T 2 ≤ ≤ 1 T 2 k k k − k k − k and is therefore closely connected to the notion of pseudospectrum (near 1) [Dav]. If A is normal, then by spectral calculus 1 z 2 1 z 2 g A , gess A . ( ) = min | − |2 ( ) = min | − |2 z σ(A) 1+ z z σess(A) 1+ z ∈ | | ∈ | | If A = V is unitary, then

g(V )=1 max σ( e(V )) = dist(1, σ( e(V ))) , gess(V ) = dist(1, σess( e(V ))) . (34) − ℜ ℜ ℜ This applies, in particular, to V (T ) so that g(V (T )) provides a good way to measure the gap at 1. Using this notation, one has for a J-unitary T that

1 T is S -Fredholm min gess(V (z T )) > 0 . ⇐⇒ z S1 ∈

Next let us show that g(V (T )) and gess(V (T )) can also be calculated more directly from T , namely g(T ) and gess(T ) as deﬁned in (32).

Proposition 26 For any J-unitary T ,

g(V (T )) = g(T ) , gess(V (T )) = gess(T ) , and 1 1 g(T ) = g(T ∗) = g(T − ) , gess(T ) = gess(T ∗) = gess(T − ) .

Proof. One can focus on g because gess is dealt with in the same way. Let us begin with an obvious reformulation of (34):

g(V (T )) = 1 max σ( e(V (T ))) . − ℜ Using Proposition 21 one now ﬁnds:

1 g(V (T )) = 2 max σ (1 + T ) 1 + T ∗T − (1 + T )∗ . − 25 1 If one sets A =(1 + T )(1 + T ∗T )− 2 , then on the r.h.s. intervenes the spectrum σ(AA∗), which except for the point 0 is equal to the spectrum σ(A∗A). Therefore

1 1 2 2 g(V (T )) = 2 max σ 1 + T ∗T − (1 + T )∗(1 + T ) 1 + T ∗T − − 1 1 2 2 = min σ 1 +T ∗T − (1 T )∗(1 T ) 1 + T ∗T − . − − Hence comparing again with the deﬁnition (32) the ﬁrst claim follows. For the others, let us conjugate the inequality in (33) with J and then T :

1 1 1 1 g 1 + T ∗T (T 1)∗(T 1) g 1 + T − (T − )∗ (T − 1)(T − 1)∗ ≤ − − ⇐⇒ ≤ − − g 1 + T T ∗ (T 1)(T 1)∗ . ⇐⇒ ≤ − − From this the last claims all follow. ✷

Corollary 3 For any J-unitary T U( ,J), one has ∈ K

T FU( ,J) inf gess(z T ) > 0 . ∈ K ⇐⇒ z S1 ∈

Due to the importance of the gaps g(T ) and gess(T ), let conclude this section with a stability analysis for them. Actually, such an analysis is even possible for g(A) and gess(A) and a general operator A on a Hilbert space, but for J-unitaries the estimates can be considerably improved.

Proposition 27 Let T FU( ,J) and K B( ) be a J-anti-selfadjoint operator, namely ∈ K ∈ K satisfying JK = K∗J. Then − K K (i) If K is compact, then T e FU( ,J) and gess(T e )= gess(T ). ∈ K (ii) If g(T ) > 6 T K , then T eK FU( ,J) with g(T eK) g(T ) 6 T K . k kk k ∈K K ≥K − k kk k (iii) If gess(T ) > 6 T K , then T e FU( ,J) with gess(T e ) gess(T ) 6 T K . k kk k ∈ K ≥ − k kk k Proof. The claim (i) follows immediately from general principles, but also from the argument below. Let us use the notations a b α β T = , eK = 1 + , c d γ δ so that a b aα + bγ aβ + bδ a + aα + bγ b + aβ + bδ T eK = + = . c d cα + dγ cβ + dδ c + cα + dγ d + cβ + dδ K Because T e U( ,J), it follows that the diagonal entries a′ = a + aα + bγ and d′ = d + cβ + dδ ∈ K satisfy (a′)∗a′ 1 and (d′)∗d′ 1 and are therefore, in particular, invertible. It follows that ≥ ≥ 1 1 1 1 1 1 K (a− )∗ (1 + α + a− bγ)− ∗ (b + aα + bδ)(1 + d− cβ + δ)− d− V (T e ) = 1 1 1 1 1 1 (1 + d− cβ + δ)− d− (c + cα + dγ) (1 + d− cβ + δ)− d− − 1 1 1 1 ((a′)− )∗(α + a− bγ)∗ (aα a (a∗)− )(d′)− = V (T ) + −1 1 − 1− 1 , (35) (d′)− (cβ + dδ)d− c cα dγ (d− cβ + δ)(d′)− − − − 26 1 1 where for the upper right entry in the second equation the identity bd− c = a (a∗)− was used. From this one can also readily write out a formula for e(V (T eK)) e(V (T−)). ℜ −ℜ Now let us suppose that K is compact. Then also its entries α, β, γ and δ are compact. Thus (35) implies immediately that V (T eK) V (T ) is compact. But now Weyl’s theorem implies that − the essential spectra are the same. Now let us bound V (T eK) V (T ). Thus one needs to control the norms of the entries of (35). − 1 Clearly the four entries of T and K are bounded by T and K respectively. As (a′)− 1 1 1 k k k 1k k k ≤ and (a′)− 1, and by Proposition 8 also a− b < 1 and d− c < 1, it follows that k k ≤ k k k k 1 1 1 1 ((a′)− )∗(α + a− bγ)∗ 2 K , (d′)− (cβ + dδ)d− c cα dγ 4 T K , k k ≤ k k k − − k ≤ k kk k and similarly

1 1 1 1 (aα a (a∗)− )(d′)− 3 T K , (d− cβ + δ)(d′)− 2 K . k − − k ≤ k kk k k k ≤ k k As 1 T , the claimed bound (ii) now follows from Lemma 3 below which holds for arbitrary block≤k operatorsk and is stated without proof (see, e.g. [AI]). For (iii) one, moreover, needs to appeal to a standard argument with singular Weyl sequences. ✷

Lemma 3 For T B( ), one has ∈ K

a b 1 1 T = > 0 a> 0 , d> 0 , c = b∗ , a− 2 bd− 2 < 1 . c d ⇐⇒ k k 5 Essentially S1-gapped J-unitaries

5.1 Deﬁnition and basic properties

1 h h Let us introduce the notation S = z C e− z e for a closed annulus of width h 0. h { ∈ | ≤| | ≤ } ≥ 1 1 Deﬁnition 6 A J-unitary T is said to be essentially S -gapped if and only if σess(T ) S = , and ∩ ∅ strictly S1-gapped if σ(T ) S1 = . The set of all essentially S1-gapped J-unitaries is denoted by GU ∩ ∅ S1 S1 ( ,J). Moreover, T is said to have an essential -gap h(T ) > 0 if the closed annulus h(T ) containsK no essential spectrum of T and there is no spectrum on the boundary of the annulus.

Note that if n = dim( ) < , all J-unitaries are essentially S1-gapped. An alternative terminology would be to callH essentially∞ S1-gaped J-unitaries essentially hyperbolic, and strictly S1-gapped J-unitaries hyperbolic, but this will not be used here as we prefer to stick with the notion linked to spectral theory.

Proposition 28 Any S1-gapped J-unitary is also S1-Fredholm, that is GU( ,J) FU( ,J). K ⊂ K The set of S1-gapped J-unitaries GU( ,J) is open in the norm topology. K

27 Proof. The ﬁrst claim results from general principles (e.g. [EE]), the second one follows directly from Theorem I.3.1 of [GK]. ✷ A natural question about essentially S1-gapped J-unitaries is whether this property is stable under compact perturbations, namely whether a Weyl type stability theorem holds as for selfadjoint and unitary operators. A positive result in this respect is the following.

Proposition 29 Let T GU( ,J) and suppose that there is a path in the resolvent set from 0 to S1. Then, if S U( ∈,J) isK such that S T is a compact operator, then S GU( ,J). ∈ K − ∈ K Proof. As S is bounded, 0 is always in its resolvent set. Therefore the claim follows immediately from analytic Fredholm theory as reviewed in Appendix D. ✷ Thus remains open the situation when there is no path in the resolvent set from 0 to S1. Again by analytic Fredholm theory, see Appendix D, it is possible that the component of the resolvent set containing S1 is ﬁlled with point spectrum after a compact perturbation. That precisely this can indeed happen is shown by example in Section 6.4.

5.2 The signature of essentially S1-gapped operators

Recall that, given T GU( ,J) and h> 0 sufficiently small, = denotes the finite dimensional subspace consisting of∈ the spanK of generalized eigenspaces for eigenvaluesE of T lying in the annulus S1 h. For this collection of eigenvalues and generalized eigenspace =, all the notations and results of Section 3.3 apply. If h is chosen sufficiently small, one canE assure that the spectrum of T S1 in h lies only on the unit circle (which does not necessarily mean that T = is unitary because E there may be non-diagonal Jordan blocks). However, this will not be assumed| because we will be interested in paths of essentially S1-gapped J-unitary operators along which eigenvalues may S1 leave or enter the unit circle, and also h.

Deﬁnition 7 The signature Sig(T ) of T GU( ,J) is the signature of , namely ∈ K E=

Sig(T ) = ν+( =) ν ( =) . E − − E By Proposition 16, the signature can also be expressed in terms of the inertia ν (λ) of the ± eigenvalues λ S1 : ∈ h Sig(T ) = ν+(λ) ν (λ) . − − λ S1 σ(T ) λ S1 σ(T ) ∈ Xh∩ ∈ Xh∩ The following example shows that in infinite dimension any value of the signature is atteined. Example Let = ( CN ) where is an infinite dimensional Hilbert space. Let the fundamental symmetryK H be ⊕ given⊕ by HJ =(1 H1) ( 1). Then for λ C with λ < 1 and η R, 1 ⊕ ⊕ − ∈ | | ∈ the operator T =(λ 1) (eıη1) (λ− 1) is essentially S1-gapped and satisfies Sig(T )= N. ⊕ ⊕ ⋄

Theorem 8 The signature is well-deﬁned (independent of the choice of h) and a homotopy invariant on GU( ,J). K

28 Proof. Changing h in a continuous manner (such that there is only discrete spectrum of T in S1 S1 h, of course) may lead to more or less eigenvalues of T in h. However, these eigenvalues always 1 come in pairs (λ, λ− ) oﬀ the unit circle with dim( ) = dim( −1 ) by Corollary 1. Moreover, by Eλ Eλ Propostion 13, the subspace + −1 is non-degenerate and has inertia (dim( ), dim( ), 0). Eλ Eλ Eλ Eλ Hence this subspace and hence this eigenvalue pair do not contribute to the the signature Sig(T ) anyway. Next let us consider a path Tt in GU( ,J). Again eigenvalues may leave the annulus S1 K h (now of ﬁxed size), but the above argument transposes directly to this case. Concerning bifurcations of eigenvalues on the unit circle it was already shown in Proposition 17 that the inertia are constant (which is a local statement) and therefore also their contribution to the signature is constant. ✷

Proposition 30 If dim( ) < , then Sig(T )=0 for all T GU( ,J)= FU( ,J)= U( ,J). H ∞ ∈ K K K Proof. Because one can choose h such that it contains the whole (only discrete) spectrum of T , it follows that = and therefore Sig(T )= Sig( ) = dim( ) dim( )=0. ✷ E= K K H − H Actually, it is not an easy endeaver to produce examples with non-trivial signature. This is the main object of Section 6.8. Let us ﬁrst provide an alternative way to calculate Sig(T ).

5.3 The spectral ﬂow of essentially S1-gapped operators

ıt It follows from the explicit formula (28) for V (e− T ) that the path

ıt ΓT = (V (e− T ))t [0,2π) , (36) ∈ in U( ) is real analytic. Therefore all results of analytic perturbation theory [Kat] apply and K assure that all discrete eigenvalues depend real analytically on t (if adequate branches are chosen at level crossings). Moreover, if T FU( ,J), then 1 is not in the essential spectrum of ıt ∈ K V (e− T ) for all t [0, 2π). Therefore it is possible to deﬁne an associated intersection index as ∈ in Section 4.2 provided that 1 is not a constant eigenvalue of ΓT . If 1 is a constant eigenvalue, then Proposition 20 implies that each point on S1 is an eigenvalue of T . As already pointed out, this may happen as shows the example in Section 6.4. On the other hand, for essentially S1-gapped J-unitaries this cannot happen, namely one has the following:

Proposition 31 Let T FU( ,J). The path Γ deﬁned in (36) is transversal in the sense that ∈ K T there is no constant eigenvalue equal to 1 if and only if T GU( ,J). ∈ K

Therefore SF(ΓT ) is well-defined for T GU( ,J). Of course, the spectral flow is just a particular case of the Bott-Maslov index and∈ the intersectiK on index defined in Sections 2.4 and 4.2: ıt [ıt SF(ΓT ) = IN (e− T )t [0,2π), 1 = BM (e− T Ψ0)t [0,2π), Ψ0 . ∈ ∈ Therefore, it counts the number of eigenvalues of T on the unit circle, but weighted by an b b orientation. Let us first note that the homotopy invariance of the spectral flow (in the sense of Section 2.4 immediately implies the following.

29 Proposition 32 If s T is a continuous path in GU( ,J), then s SF(Γ ) is constant, 7→ s K 7→ Ts namely SF(ΓT ) is a homotopy invariant.

Furthermore, let us provide an explicit proof of the fact that SF(ΓT ) is trivial in finite dimensions, even though this also follows from Proposition 30 combined with Theorem 9 below. Proposition 33 If n = dim( ) < , then SF(Γ )=0 for all T GU( ,J). H ∞ T ∈ K Proof. The unitaries V (z T ) U(2n) are traceclass. Therefore, the spectral flow by 1 can be calculated as the total winding∈ number: 2π dt ıt ıt SF(Γ ) = Tr V (e− T )∗∂ V (e− T ) . T 2πı t Z0 Now by Proposition 20 one finds ıt 1 1 ıt 1 ıt ıt e− a− c∗(d− )∗ e (a∗)− 0 Tr V (e− T )∗∂tV (e− T ) = ı Tr 1 −ıt 1 − ıt 1 (d− )∗b∗ e (d− )∗ 0 e− d− 1 1 = ı Tr (a∗a)− +(dd∗)− − But as a∗a = 1 + c∗c and dd∗ = 1 + cc∗, the unitary invariance of the trace implies that the last expression vanishes so that SF(ΓT )=0. ✷ ıt ıt In general, the operator V (e− T )∗∂tV (e− T ) is not trace class so that the above calculation cannot be carried through. It may not come as a surprise that this orientation is given by the inertia of the eigenvalue. Actually, this follows from an explicit formula proved next. Proposition 34 Let T FU( ,J) have a simple unit eigenvalue eıϕ with eigenvector φ. Let ıθt ∈ K ıt e be a discrete non-degenerate eigenvalue of V (e− T ) depending analytically on t and satisfying θ0 = ϕ. Then ∂ θ = φ∗J φ . t t t=0 − Proof. By considering e ıϕT instead of T , one sees that it is sufficient to consider the case − ϕ =0. Thus one has Tφ = φ as well as JT ∗Jφ = φ and V (T )φ = φ. The phase speed can now be calculated using the quadratic from Q1(T ) defined in (30): 1 ıt (a∗)− 0 ∂t θt = φ∗Q1(T ) φ = ıφ∗∂tV (e− T ) φ = φ∗ − 1 φ . t=0 − t=0 0 d−

Now one can proceed as follows: 1 (a∗)− 0 0 0 ∂t θt = φ∗ − JT ∗Jφ + φ∗JTJ 1 φ t=0 0 0 0 d− 1 (a∗)− 0 a∗ c∗ a b 0 0 = φ∗ − − φ + φ∗ − 1 φ = φ∗J φ , 0 0 b∗ d∗ c d 0 d− − − − 1 1 where in the last equality the relation (a∗)− c∗ = bd− was used which is equivalent to the relation c∗d = a∗b, see Propostion 8. ✷ Based on the previous proposition, one can now prove that the spectral ﬂow and the signature are equal, up to a sign.

30 Theorem 9 For all T GU( ,J), one has Sig(T )= SF(Γ ). ∈ K − T

Proof. Because both Sig(T ) and SF(ΓT ) are homotopy invariants by Theorem 8 and Proposi- tion 32, it is possible to deform T in such a manner that all eigenvalues on S1 are simple. But ıt for such T the orientation of the spectral ﬂow of the eigenvalues of V (e− T ) through 1 is due to Proposition 34 given by the inertia of the corresponding eigenvalue of T multiplied by 1. Summing over all inertia is thus equal to minus the spectral ﬂow. −✷

5.4 Block diagonalization of J-unitaries Let us begin with the construction of a number of objects naturally associated to a S1-gapped J-unitary T with a given (and ﬁxed) gap h(T ). First of all, the spectrum can be split in three subsets: h(T ) h(T ) ∆ = z σ(T ) z e , ∆ = z σ(T ) z e− , > { ∈ || | ≥ } < { ∈ || | ≤ } S1 and ∆= = σ(∆) (∆+ ∆ ) h T . Of course, this splitting depends on the choice of \ ∪ − ⊂ ( ) h(T ), but this will be irrelevant for our purposes below. Associated to the splitting are Riesz projections denoted by P> = P∆> , P< = P∆< and P= = P∆= , as well as T -invariant subspaces by = Ran(P ), = Ran(P ) and = Ran(P ) and T ∗-invariant subspaces = Ker(P ), E> > E< < E= = F> > = Ker(P ) and = Ker(P ). If necessary, also the notation (T )= , etc., will be used. F< < F= = E< E< Let us also use P = P> + P= and P = P< + P= as well as and . Frames for <, =, ≥ ≤ E≥ E≤ E E > and so on will be denoted by Φ<, Φ=, Φ . Then Φ = (Φ<, Φ=) is also a frame for . Of E 1 ≥ ≤ E≤ course, for a strictly S -gapped operator T , one has P= =0 and Φ= =0. From general principles (see Appendix C),

1 = P< + P= + P> = P< + P , 0 = P

, (37) ≥ ≥ ≥ as well as related identities. Every v < in the range of the idempotent P< satisﬁes P

< = , < = , (38) E F≥ F E≥ as well as again related identities. Moreover, it follows from Proposition 11 that

P<∗ = JP> J, P=∗ = JP= J. (39) This implies = J ⊥ , = J ⊥ , ⊥ = J . (40) E< F> E> F< F= E= Finally, Proposition 12 shows that < and >. In particular, < and > are isotropic ≤ ≥ (and Lagrangian for a strictly S1-gappedE ⊥EJ-unitary).E ⊥E Thus E E b b Φ<∗ J Φ = 0 , Φ∗ J Φ> = 0 . (41) ≥ ≤

Another fact that will be of importance later on is that = is non-degenerate by Proposition 13. As a ﬁnite-dimensional space is also closed and thereforeE Proposition 4 combined with = E= K ( < + >)+ = implies that E E E b ( =)⊥ = < + > . (42) b E E E 31 b b b Next it follows from Proposition 1(ii) that ( =+ <)⊥ =( =)⊥ ( <)⊥, so that the equality (42) b b E E E ∩ E shows ( + )⊥ =( + ) ( )⊥ = . From this and a similar argument one deduces E= E< E< E> ∩ E< E< b b b ( )⊥ = + , ( )⊥ = + . b E< E= E< E> E= E> Finally let us note that there are also connections between the spectral subspaces of T and T , b b ∗ denoted by (T ), (T ∗), etc., for now. Indeed, E< E< (T ∗) = Ran(P ∗ ) = Ker(P )⊥ = (T )⊥ = J (T ) , (T ∗) = J (T ) . E< < < F< E> E> E<

Proposition 35 Let T GU( ,J). Then and form a Fredholm pair. ∈ K E< E> Proof. As , and + are all ranges of Riesz projections, they are closed (see Proposi- E< E> E< E> tion 42 in Appendix B). Moreover, < > = 0 and dim(( < + >)⊥) = dim( =) < so that all conditions of Deﬁnition 10 in AppendixE ∩E B are{ } satisﬁed. E E E ∞ ✷

Associated to the Fredholm pair < and > is always a Fredholm index, which in the present situation is Ind( , )= codim( E+ ).E This index is of little interest in the present situation E< E> − E< E> though. For example, it does depend on the choice of the gap h(T ) because this may modify the appearing codimension. The signature deﬁned in Section 5.2 below is a more interesting and topological index. The main consequence of the Fredholm property relevant in the present context is the following. Proposition 36 Let T GU( ,J). Then ∈ K 1 1 − − P< = Φ< Φ>∗ J Φ< Φ>∗ J, P> = Φ< Φ<∗ J Φ> Φ<∗ J, and 1 − P= = Φ= Φ=∗ J Φ= Φ=∗ J.

Proof. The oblique projection P< has by definition range < and kernel <. But < = by E F F E≥ (38) and and > only differ by a finite dimensional subspace, it follows that < and < form E≥ E E F a Fredholm pair. Moreover, = 0 and + = by (37). Thus one can appeal to E< ∩F< { } E< F< K Proposition 41 to write out P . Now Φ is a frame for by definition. Because ⊥ = J < < E< F< E> by (40), a frame for ⊥ is given by JΦ . Thus Proposition 41 allows to deduce the formula for F< > P . The one for P follows in a similar manner. For the last formula, let us note that is < > E= non-degenerate by Proposition 13. Therefore the matrix Φ=∗ JΦ= is invertible. Hence the formula is well-defined and obviously an idempotent with range =. As =⊥ = J = by (40), the kernel of the r.h.s. is indeed as desired. E F E ✷ F= Finally let us come to a block diagonalization of T GU( ,J). For that purpose, let us set ∈ K M = (Φ<, Φ=, Φ>) . (43) One can check explicitly using (41) that

0 0 Φ>∗ JΦ< 0 0 Φ>∗ JΦ< M ∗ J M = 0 Φ∗ J Φ 0 = 0 Φ∗ J Φ 0 .  = =   = =  Φ<∗ JΦ> 0 0 (Φ>∗ JΦ<)∗ 0 0     32 Thus the inverse of M is given by 1 0 0 (Φ>∗ JΦ<)− 1 1 M − = 0 (Φ=∗ J Φ=)− 0 M ∗ J,  1  (Φ<∗ JΦ>)− 0 0   where the matrix is in the grading of (43). The invertibility of the three operators follows as in Proposition 36. Now the spaces , and are invariant for T . Therefore the span of T Φ E< E= E> < lies in < and thus (41) implies Φ∗ JT Φ< = 0. Similarly Φ∗ JT Φ> = 0. Therefore one readily checksE that ≤ ≥ 1 (Φ>∗ JΦ<)− Φ>∗ J T Φ< 0 0 1 1 M − T M = 0 (Φ=∗ JΦ=)− Φ=∗ J T Φ= 0 .  1  0 0 (Φ<∗ JΦ>)− Φ<∗ J T Φ>   But as T Φ< = Φ<Φ<∗ T Φ< by the invariance property and because Φ<Φ<∗ is the orthogonal projection on , and similarly for T Φ and T Φ , it follows that E< > = Φ<∗ T Φ< 0 0 1 M − T M = 0 Φ∗ T Φ 0 . (44)  = =  0 0 Φ>∗ T Φ>   The three entries are the restrictions of T to the invariant subspaces <, = and >. Hence their spectra lie inside, on or respectively outside the unit disc. One canE nowE furtherE choose the frames such that the diagonal blocks become Jordan blocks, in particular, the ﬁnite dimensional S1 matrix Φ=∗ T Φ=. It is also possible to use formula (44) to construct -gapped J-unitary operators from a disjoint decomposition = + + into two isotropic subspaces and and a K E< E= E> E< E> non-degenerate subspace . E= 6 Examples

The aim of this section is to illustrate the concepts and results of the paper by some examples. The formulas are often a bit easier if one works with another fundamental symmetry G obtained from J by the Cayley transform C: 0 ı 1 1 1 ı 1 G = C∗JC = − , C = − . ı 1 0 √2 1 ı 1 One then has G-Lagrangian subspaces, G-unitaries, FU( ,G), etc., and the passage from one to the other is just the Cayley transform. Note that G is theK standard symplectic form, multiplied by a complex unit in order to make it self-adjoint and involutive. Symplectic matrices and transfer operators (see Section 6.6 below) are usually given in a form that is G-unitary rather than J-unitary, so that, pending on the scientiﬁc background, the reader may be more familiar with it. Another reason why to prefer one or the other representation in concrete situations is the following fact (which can be deduced from the examples in Section 6.3): J-unitary unitaries may be diagonal, but not G-unitaries; on the other hand, G-unitaries with spectrum oﬀ the unit circle can be diagonal, but not J-unitaries. Hence one can diagonalize unitaries within the group of J-unitaries, and hyperbolic operators within the group of G-unitaries.

33 6.1 Positivity in linear Hamiltonian systems In this section, let us consider a Hamiltonian system of the form

J ∂ T E = ı (H + EP ) T E , T E FU( , J, 1) . (45) t t t t t 0 ∈ K R Here E is a spectral parameter, and Ht = Ht∗ and Pt 0 are Lipshitz continuous functions B ∈ E E ≥ E in ( ). As one readily checks that ∂t (Tt )∗JTt = 0 and as the initial condition T0 is J- K E unitary, it follows that the solution Tt of (45) is also J-unitary. Because the initial condition E is, moreover, a 1-Fredholm operator and the set of these operators is open, the solution Tt is also 1-Fredholm, at least for sufficiently small t. For fixed (small) t, it is also possible to vary E FU E in compact intervals such that still Tt ( , J, 1). Therefore there is a two-parameter E FU ∈ K family Tt in ( , J, 1) associated to (45) and one can study the intersection index defined in K E Section 4.2 via the spectral flow of the unitaries V (Tt ). The transversality in t and E follows from the following positivity results, which are also crucial in the context of Sturm-Liouville type oscillation theory associated to the Hamiltonian system. In the finite-dimensional case, (i) and (ii) are due to Lidskii [Lid], and (iii) to Bott [Bot]. Similar monotonicity results hold for the E unitaries πΨ(Tt Φ) whenever Φ and Ψ form a Fredholm pair of Lagrangian subspaces. Theorem 10 The solution of the Hamiltonian system (45) satisfies the following. 1 E E (i) If Ht + EPt > 0, then ı V (Tt )∗∂tV (Tt ) > 0. 1 E E (ii) If Ht + EPt < 0, then ı V (Tt )∗∂tV (Tt ) < 0. E 1 E E (iii) If ∂ T =0, then V (T )∗∂ V (T ) 0. E 0 ı t E t ≥ E E E Proof. Recall that V (T )= π(Ψ )∗ π(T Ψ )= π(Ψ )∗ T π(Ψ ) so that t 0 t 0 0 t · 0 1 E E 1 E E V (T )∗∂ V (T b) = c bT π(Ψb) ∗∂cT bπ(Ψ ) ∗ ı t t b t b ı t · b 0 t tb· 0 ∗ 1 c π(Ψ0) Ec E π(Ψ0) = A∗ b b (T )∗J ∂bTb A ı 1 t t t 1 2 b b E E b b = A∗ Ψ∗ 0 (T c)∗J ∂bT cΨ A ı 0 ⊕ t t t 0 2 E E = A∗ (bT )∗J ∂tT A, b ı t b t where in the second equality Proposition 9 was used and A is some invertible operator. Now using (45), 1 E E E E V (T )∗∂ V (T )=2 A∗ (T )∗(H + EP ) T A, ı t t t t t t t which implies (i) and (ii). Similarly,

1 E E 2 E E V (T )∗∂ V (T ) = A∗ (T )∗J ∂ T A. ı t E t ı t E t Again from (45) one ﬁnds for ǫ> 0

E E+ǫ E E+ǫ ∂s (Ts )∗ J Ts = ı ǫ (Ts )∗ Ps Ts . 34 E E E E As (Tt )∗JTt = J =(T0 )∗JT0 , one thus has

1 E E 1 E E+ǫ E E (Tt )∗ J ∂ETt = lim (ı ǫ)− (Tt )∗ J Tt (Tt )∗ J Tt ǫ 0 ı → − 1 E E+ǫ E E+ǫ E E+ǫ E E = lim (ı ǫ)− (Tt )∗ J Tt (T0 )∗ J T0 + (T0 )∗ J T0 (T0 )∗ J T0 ǫ 0 → − − t E E E E = ds (Ts )∗ Ps Ts + (T0 )∗ J ∂ET0 . Z0 Because Ps is non-negative and the last summand vanishes by hypothesis, this implies (iii). ✷

6.2 The calculation of bound states by a Bott-Maslov index Let H = H +K be a self-adjoint bounded operator on given as a sum of a self-adjoint operator 0 H H0 called the free Hamiltonian and a compact operator K called the perturbation. In typical applications, H0 has a band spectrum which is purely absolutely continuous. However, all that is needed for the below is an interval ∆ R, possibly unbounded, in the resolvent set ρ(H ) ⊂ 0 of H0. It follows from Weyl’s stability theorem for essential spectra that H has only discrete spectrum in ∆. The new eigenvalues and eigenvectors of H are called bound states. The aim of this section is to reformulate the calculation of these bound states as an intersection problem for a path of Lagrangian Fredholm pairs. With some more technical eﬀort, one can also deal with unbounded-selfadjoint operators. For that purpose, let us use the frames 1 1 E 2 1 E 2 1 Φ = (1 +(H E 1) )− 2 , Φ = (1 +(H E 1) )− 2 , H E 1 − 0 H E 1 0 − − 0 − 1 as well as Ψ= . These three frames are G-Lagrangian on . 0 K R E Proposition 37 Let E ∆ ρ(H0) . Then the Lagrangian subspaces associated to Φ0 and Ψ form a Fredholm pair∈ and⊂ their intersection∩ is empty for E ∆. Furthermore, also the Lagrangian subspaces associated to ΦE and Ψ form a Fredholm pair∈ and the dimension of the intersection is equal to the multiplicity of E as bound state of H. Proof. In order to check the Fredholm pair property, let us use the characterization of Theo- E 1 1 1 2 1 rem 2(ii), forumated for G instead of J. As Ψ∗GΦ0 =(E H0)( +(H0 E ) )− 2 is self-adjoint, it is Fredholm if and only if 0 is not in its essential spectrum.− But this− is given because E is not E even in the spectrum. Next let us suppose that Φ0 and Ψ have a non-trivial intersection, namely E that there are non-vanishing vectors v,w such that Φ0 v =Ψw. Then the upper equation of 2 1 ∈ H this identity implies (1 +(H0 E 1) )− 2 v = w and the lower one (H0 E 1)w =0. This would − − E mean that E is an eigenvalue of H0, in contradiction to the hypothesis. For the pair Φ and Ψ E E one proceeds in the same way. Alternatively, one can use the fact that Φ is obtained from Φ0 by compact perturbation. ✷ Due to this proposition, one can calculate the eigenvalues of H in ∆ (namely the bound states) via the spectral ﬂow of the unitaries

E E E ∆ V = π(CΨ)∗π(CΦ ) U( ) . ∈ 7→ ∈ H 35 By Theorem 2, this path is actually in Uess( ). Of course, π(CΨ) = 1 and by the explicit expression of ΦE: H

E 1 E E 1 V = 1 ı(H E 1) 1 + ı(H E 1) − = (G + ı 1)(G ı 1)− , (46) − − − − E 1 where G =(E 1 H)− is the resolvent whenever it exists. Precisely at the bound states this inverse does not exist,− but nevertheless V E is well-deﬁned by the ﬁrst formula.

E Proposition 38 Let ∆ ρ(H ). Then E ∆ V is a real analytic path in Uess( ) satisfying ⊂ 0 ∈ 7→ H 1 E E E (V )∗ ∂E V > 0 , lim V = 1 . ı E − | |→∞ In particular, all eigenvalues of V E rotate in the positive sense as a function of E and cross 1 transversally in the positive orientation. These latter eigenvalue crossings correspond exactly at the eigenvalues of H, with equal multiplicities.

Proof. It only remains to verify the positivity of

1 E E 1 E 1 E 1 (V )∗∂ V = (V )∗ ı 1 + ı(H E 1) − V ( ı) 1 + ı(H E 1) − ı E ı − − − − h 1 1 i = 2 1 + ı(H E 1) − ∗ 1 + ı(H E 1) − − − 1 = 2 1 +(H E 1)2 − , − which is obvious. ✷

6.3 The unitary V (T ) for 2 2 matrices × This elementary section provides the normal forms of 2 2 J-unitaries T , their G-unitary Cayley ıt × transform C∗TC, and the associated unitaries V (e− T ) Example 1 Let us consider, for ϕ, η R, ∈ η+ıϕ ıϕ cosh(η) sinh(η) e 0 T = e C∗TC = η+ıϕ . sinh(η) cosh(η) 0 e− One then ﬁnds eı(ϕ−t) ıt cosh(η) tanh(η) V (e− T ) = −ı(ϕ−t) . tanh(η) e − cosh(η) ! This matrix satisﬁes

ıt 2 cos(ϕ t) ıt Tr(V (e− T )) = − , det(V (e− T )) = 1 . cosh(η)

ıt Hence the eigenvalues of V (e− T ) are

2 ıθ±,t cos(ϕ t) cos (ϕ t) e = − ı 1 2 − . cosh(η) ± s − cosh (η)

36 In particular, 1 max e(eıθ±,t ) = , t [0,2π] ℜ cosh(η) ∈ which is thus strictly less than 1. Hence there is a gap to the value 1 on the circle. ⋄ Example 2 Now let us consider

ıη ıϕ e− 0 ıϕ cos(η) sin(η) T = e , C∗TC = e − . 0 eıη sin(η) cos(η) Then ı(η t ϕ) ıt e − − 0 V (e− T ) = ı(η t+ϕ) . 0 e− − Hence the spectrum can immediately be read oﬀ. ⋄ Example 3 Let us consider the Jordan block

ıϕ 1 ıa ıa ıϕ 1 2a T = e , C∗TC = e , −ıa 1+ ıa 0− 1 − where a R. Then one ﬁnds ∈ eı(ϕ−t) ıa ıt 1+ıa 1+ıa V (e− T ) = ıa e−ı(ϕ−t) . 1+ıa 1+ıa !

This matrix satisﬁes

ıt 2 cos(ϕ t) ıt 1 ıa Tr(V (e− T )) = − , det(V (e− T )) = − . 1+ ıa 1+ ıa Hence the eigenvalues are

1 eıθ±,t = cos(ϕ t) ı a2 + sin2(ϕ t) . 1+ ıa − ± − q In particular, for t = ϕ and say a> 0, one has eıθ+,ϕ =1 and e(eıθ−,ϕ )=1 a2. Hence there is ℜ − a gap to the value 1 on the circle. Furthermore,

1 a + 1 m(eıθ+,t ) = a (t ϕ)2 + + ((t ϕ)3) , ℑ 2 1+ a2 − O − that is, the eigenvalue is touching 1 only from above. Therefore, there is no spectral ﬂow of ıt t V (e− T ) through 1. 7→ ⋄ Example 4 Finally, let us study the family

E E ı E E 2 E 1 T = − , C∗T C = − . E E + ı 1 0

37 Then e−ıt E ıt E+ı E+ı V (e− T ) = E eıt . E−+ı E+ı ! Thus ıt 2 cos(t) ıt E ı Tr(V (e− T )) = , det(V (e− T )) = − , E + ı E + ı so that 1 eıθ±,t = cos(t) ı E2 + sin2(t) . E + ı ± q Let us now focus on E > 0. Then E cos(t)+ E2 + sin2(t) e eıθ+,t , e eıθ−,t . ( ) = 2 ( ) 0 ℜ 1+pE ℜ ≤

If E < 1, then cos(t) = E has two solutions t+ > 0 and t < 0 for which then θ+,t± = 0. For − E =1 there is one solution t =0, while for E > 1 there is no solution and hence e(eıθ+,t ) < 1 uniformly in t. This is a bifurcation, but the spectral ﬂow remains invariant. ℜ

6.4 A ﬁnite rank perturbation of an S1-gapped G-unitary

As motivation to this section, let us recall some results about the operator family St = S + tQ 2 on ℓ (Z) where S is the shift and Q = 0 1 . For t = 0 it is the shift, but S1 = S = S+ S − is the sum of a uni-lateral shift and its| adjoint,ih | where the direct sum decomposition is ℓ2(⊕Z) = 2 Z 2 Z Z N Z Z Z ℓ ( +) ℓ ( ) with + = = 1, 2,... and = + = 0, 1, 2,... , and S = S ℓ2(Z±). ⊕ − { } − \ { − − } ± n | Thus the spectrum of S1 is the unit disc ﬁlled with point spectrum because n 1 λ n is an eigenvector with eigenvalue λ C as long as λ < 1. As S is a ﬁnite rank perturbation≥ | i of S, ∈ | | 1 this provides an example for the scenario (ii) in the analytic Fredholm theoremP of Appendix D. Note that the Fredholm index is Ind(S1)=1 1=0. On the other hand, the spectrum of St for t [0, 1) is only the unit circle (see e.g. [Hal,− Problem 102], or argue as below). On the other ∈ hand, the family St is norm-continuous in t. This example shows that the spectrum as a set is only upper semi-continuous, namely lim supt 1 σ(St) σ(S1), but this inclusion may be strict. → ⊂ The aim of this section is to construct an example of Tt showing that the same phenomena may happen within the class of G-unitary operators (it is impossible in the class of normal operators). Let still S be the two-sided shift on ℓ2(Z) and r (0, 1). Then one starts from ∈ rS 0 T0 = 1 , 0 r− S acting on the Hilbert space ℓ2(Z) C2. The operator T is G-unitary and its spectrum is ⊗ 0 1 1 1 σ(T ) = rS r− S , 0 ∪ 1 so that T0 is clearly S -gapped. Now let P+ = 1 1 and P = 0 0 be the one-dimensional | ih | − | ih | projections in ℓ2(Z) onto the subspaces spanned by 1 and 0 respectively and then set | i | i 0 P K = + . P 0 − + 38 One has GK = K∗G so that K is in the Lie algebra of the G-unitaries and thus − 1 P + cos(t)P sin(t)P etK = − + + + , sin(t)P 1 P + cos(t)P − + − + + is G-unitary and a compact perturbation of 1. Let us set 1 tK rS( P+)+ r cos(t)SP+ r sin(t)SP+ Tt = T0 e = − 1 1 1 . (47) r− sin(t)SP r− S(1 P )+ r− cos(t)SP − + − + + Claim σ(T )= σ(T ) for t = π , 3π , and t 0 6 2 2 1 σ(T π ) = σ(T 3π ) = λ C r λ r− . (48) 2 2 { ∈ | ≤| | ≤ }

Furthermore, in agreement with analytic Fredholm theory one has σdis(T π )= σ(T π ) σ(T0). 2 2 \ This shows that the essentially S1-gapped operators are not stable under compact perturbations. Let us begin the proof of the claim by rewriting (47). First recall that 1 1 1 1 S+∗ S+ = P+ , S+S+∗ = , S S∗ = P , S∗ S = . − − − − − − − Furthermore, with the partial isometry Q = 0 1 as above, | ih |

Q∗Q = P+ , QQ∗ = P , Q = SP+ = P Q. − − 2 2 For the matrix entries of Tt one ﬁnds in the grading of ℓ (Z+) ℓ (Z ) ⊕ − 0 0 S 0 SP = , S(1 P ) = + . + Q 0 − + 0 S − 2 2 2 2 Therefore Tt acting on ℓ (Z+) ℓ (Z ) ℓ (Z+) ℓ (Z ) becomes ⊕ − ⊕ ⊕ −

rS+ 0 0 0 r cos(t)Q rS r sin(t)Q 0 Tt =  − 1  . 0 0 r− S+ 0 1 1 1  r− sin(t)Q 0 r− cos(t)Q r− S  − −   Apart from t =0, the points t = π , 3π are particularly simple because the 4 4 matrix operator 2 2 × decouples in two 2 2 matrix operators. Let us start with ×

rS+ 0 0 0 0 rS r Q 0 T π =  − 1  . 2 0 0 r− S+ 0 1 1  r− Q 0 0 r− S  − −   2 Hence it is suﬃcient to carry out the spectral analysis of the following block operators on ℓ (Z+) ℓ2(Z ): ⊕ − 1 rS+ 0 r− S+ 0 A+ = 1 1 , A = . r− Q r− S − rQ rS − − − 39 Indeed, then σ(T π )= σ(A ) σ(A ). Note that neither A nor A are G-unitary (for any G). 2 + + ∪ − − 1 1 1 As S are unilaterl shifts, a singular Weyl sequence argument shows that rS r− S σ(A ). ± ± Let us focus on A and look at the trial state ∩ ⊂ − n n ψ = α λ+ n β λ| | n , (49) | i ⊕ − | i n 1 n 0 X≥ X≤ 2 2 where λ < 1 and α, β C. Clearly ψ ℓ (Z+) ℓ (Z ). As Q is a perturbation acting ± − non-trivially| | only on the subspace∈ spanned∈ by 1 , any⊕ eigenfunction of A has to be of this | i − form. Furthermore

1 n 1 n A ψ = αr− λ+ λ+ n αrλ+ 0 + β r(λ )− λ| | n . − | i ⊕ | i − − | i n 1 n 1 X≥ X≤− 1 Now for a given λ C with λ (r, r− ) let us choose λ , α and β such that ∈ | | ∈ ± λ r rλ λ = + = , α = 1 , β = + = r2 . r λ λ − 1 Then A ψ = λψ. Thus the annulus λ C r < λ < r− is in the point spectrum of A so − − that σ(A ) contains the closure of the{ annulus.∈ | Outside| | of} the annulus one could have point − spectrum (this is the only spectrum allowed by analytic Fredholm theory because A is bounded − and invertible), but actually the above argument shows that there are no further eigenvalues outside of the annulus.

One can attempt to apply a similar argument to A+:

n 1 1 n A+ψ = αrλ+ λ+ n α( r− )λ+ 0 + β (rλ )− λ| | n . | i ⊕ − | i − − | i n 1 n 1 X≥ X≤− 1 But now rλ+ = (rλ )− is incompatible with λ < 1 and r < 1. Hence A+ has no eigenvalue S1 − 1S1 | ±| π and σ(A+) = r r− . Combining the above one thus concludes that (48) holds for t = 2 . 3π ∪ The case t = 2 is dealt with similarly. π 3π Next let us do the spectral analysis of Tt for t = 2 , 2 . By analytic Fredholm theory as described in Appendix D either σ(T ) σ(T ) only contains6 discrete spectrum, or all points in t \ 0 this set are eigenvalues. Thus let us look for eigenvalues of Tt. The eigenvectors have to be of the form

n n n n ψ = α λ+, n, + β λ| ,| n, + α λ+, n, + β λ| ,| n, , ↑ ↑| ↑i ↑ − ↑| ↑i ↓ ↓| ↓i ↓ − ↓| ↓i n 1 n 0 n 1 n 0 X≥ X≤ X≥ X≤ where all λ’s are of modulus less than 1 and the coeﬃcients α and β are complex numbers, and the states n, and n, correspond to the ﬁrst and second component of ℓ2(Z) ℓ2(Z). | ↑i | ↓i ⊕ Applying Tt now gives

n β r n α λ+, n β n Ttψ = α rλ+, λ+, n, + ↑ λ| ,| n, + ↓ ↓ λ+, n, + ↓ λ| ,| n, ↑ ↑ ↑| ↑i λ , − ↑| ↑i r ↓| ↓i rλ , − ↓| ↓i n 1 n 0 n 1 n 0 X≥ − ↑ X≤ X≥ − ↓ X≤ 1 + r(α cos(t)λ+, + α sin(t)λ+, ) 0, + (α cos(t)λ+, α sin(t)λ+, ) 0, . ↑ ↑ ↓ ↓ | ↑i r ↓ ↓ − ↑ ↑ | ↓i 40 From this equation one deduces with some care that there cannot be any eigenvectors in the 1 π 3π annulus λ C r< λ

1 ıt cosh(h)S sinh(h)S tK P+ + e P+ 0 C T0C∗ = , Ce C∗ = − ıt . sinh(h)S cosh(h)S 0 1 P + e− P − + +

From this the J-unitary CTtC∗ can readily be written out. For (28), one needs the inverse of the adjoint of the upper left as well as the lower right corner of CTtC∗, namely

ıt 1 ıt cosh(h)(1 P + e− P )S∗ − = sech(h)S(1 P + e P ) , − + + − + + and ıt 1 ıt cosh(h)S(1 P + e− P ) − = sech(h)(1 P + e P )S∗ . − + + − + + Thus ıt z sech(h)S(1 P+ + e P+) tanh(h) V (z Tt) = − 2ıt ıt . tanh(h)(P e P 1) z sech(h)(1 P + e P )S∗ + − + − − + + In particular, z sech(h)S tanh(h) V (z T0) = . tanh(h) z sech(h)S∗ − After Fourier transform the spectral analysis of the 2 2 matrix (see Example 1 above) applies 1 × 1 and this shows that e(V (z T0)) sech(h)− 1 for all z S . Now V (z T π ) is a ﬁnite range ℜ ≤ ∈ 2 perturbation of V (z T0). Thus the essential spectrum is unchanged by Weyl’s theorem. Moreover, as every z S1 is an eigenvalue of T by the above, it follows that 1 is always an eigenvalue of ∈ V (z T π ). Note that this is not a contradiction to the the fact 0 σ(Qz(T )), because the quadratic 2 6∈ form σ(Qz(T )) has as many negative as positive eigenvalues.

6.5 A loop in FU( , G) with non-trivial intersection index K This section completes the analysis of the example in the last section by calculating the intersection number of the loop Γ=(Tt)t [0,2π) where Tt is given by (47). As a ﬁnite rank perturbation 1 ∈ of a S -gapped G-unitary T0, this is clearly a path in FU( ,G). Moreover, the analysis of the K π 3π proceeding section shows that there are two intersection along the loop, namely at t = 2 , 2 , and both are of multiplicity 1. Hence IN(Γ) could be 2, 0 or 2 and it remains to analyze the orientation at the two intersections. For that purpose,− let us ﬁrst note that

1 1 1 P+ 0 Tt∗G ∂tTt = Tt∗G Tt K = GK = 0 . ı ı ı 0 P+ ≥ It follows from Proposition 22, for the fundamental symmetry G instead of J, and the fact that the eigenfunction (49) does not vanish on the range of P+ that the phase speed is positive. Therefore IN(Γ) = 2.

41 6.6 Transfer operators This section merely presents a class of G-unitaries typically associated to discrete Schrödinger operators. More speciﬁc cases are studied in the following sections. Let H = H∗ B( ) and 1 ∈ H A B( ) such that A− B( ) exists. Then ∈ H ∈ H 1 2 HA− A∗ T = 1 − , (50) A− 0 is G-unitary on = . Let us look for an eigenvalue λ S1 and eigenvector v of T : K H ⊗ H ∈ w 1 v v 2 HA− v A∗w v T = λ 1− = λ . w w ⇐⇒ A− v w The second equation enforces v = λAw. Replacing this into the ﬁrst one, the eigenvalue equation is equivalent to searching for w with ∈ H (2 H λA (λA)∗)w = 0 , (51) − − namely for w in the kernel of the self-adjoint operator H(λ)=2 H λA (λA)∗. If such a ∈ H − − w and λ are given, then λAw λAw T = λ , w w and, if the eigenvalue λ is simple, its signature is the sign of λAw ∗ λAw G = 2 m λw∗Aw . w w − ℑ For the rest of the spectrum one deduce from a Weyl sequence argument 0 σ(H(λ)) λ σ(T ) . ∈ ⇐⇒ ∈ In concrete situations, this also allows to examine whether T is S1-gapped. Furthermore, one checks 1 1 1 (2 H ı 1 ı A∗A)A− (2 H ı 1 + ı A∗A)A− CTC∗ = − − 1 − 1 . 2 (2 H + ı 1 ı A∗A)A− (2 H + ı 1 + ı A∗A)A− − This, and consequently the formulas for V (z T ) and Qz(T ) considerably simplify if A is unitary. Thus let us assume this from now on so that 1 1 (H ı 1)A− HA− CTC∗ = − 1 1 . HA− (H + ı 1)A− Due to the general formulas, 1 1 z(H + ı 1)− A∗ H(H + ı 1)− V (z T ) = 1 1 , A(H + ı 1)− HA∗ zA(H + ı 1)− − and 2 1 2 1 A(H + 1)− A∗ zAH(H + 1)− Qz(T ) = − 2 1 − 2 1 . z(H + 1)− HA∗ (H + 1)− − The speed of the eigenvalue of V (λ T ) crossing 1 is due to Proposition 34 equal to

1 λAw ∗ λAw 2 m λw∗Aw Q (T ) = ℑ . λAw 2 + w 2 w λ w λAw 2 + w 2 k k k k k k k k

42 6.7 S1-gapped G-unitaries with discrete spectrum on the unit circle

For a ﬁrst concrete example of a transfer operator of the form (50), let A = 1 and H = H∗ B( ) ∈ H be such that σess(H) [ 1, 1] = and suppose that Hw = µw for some µ ( 1, 1) and a unit vector w . For sake∩ − of simplicity,∅ let us also suppose that there is only∈ one− such eigenvalue. ∈ H 1 1 Then λ is an eigenvalue of T if 2µ = λ + λ− . Actually then both λ and λ− are eigenvalues of T :

1 1 λw λw λ− w 1 λ− w T = λ , T = λ− . w w w w As, moreover, µ ( 1, 1), it follows that λ S1. Furthermore, a Weyl sequence argument shows ∈ − ∈ that T is essentially S1-gapped. Let us examine the signature of λ = eıϕ:

λw ∗ λw G = 2 sin(ϕ) . w w − Thus the signature is negative for λ in the upper arc of S1, and positive for λ in the lower one. As the signature is the sum over all eigenvalue pairs on the unit circle, one has Sig(T )=0. 2 A concrete example is 2H = E S S∗ cot(α) 0 0 on = ℓ (N) with E [ 1, 1] and − − − | ih | H 6∈ − S is the unilateral shift (which was also denoted by S+ in Section 6.4). This models a half-sided discrete Laplacian with adequate boundaryb b condition α [0, π]. ∈ b 6.8 Essentially S1-gapped G-unitaries with non-vanishing signature This example is inspired by the two-dimensional Harper model describing a tight-binding electron on a square lattice submitted to a magnetic field (in the Landau gauge). This connection will 2 be made more explicit elsewhere [SV]. Let again = ℓ (N) and 2H = E S S∗ where S is H − − the unilateral shift. Furthermore, the unitary is A = eıθX where X is the position operator on ℓ2(N) defined by X n = n n and finally θ [0, 2π) is a phase (the magneticb fluxb throughb the unit cell). The G-unitary| i transfer| i operator T∈depending on the two parameters E R and θ is ∈ then explicitly given by (E S S )e ıθX e ıθX T = ∗ − − . − e−ıθX − 0 − Let us look for eigenvalues λ = eıϕ of T byb analyzingb (51), which becomes more explicitly

(E S S∗ 2 cos(θX + ϕ))w = 0 . − − −

This is the Schrödinger equation hbθ,ϕwb= Ew for the one-dimensional (critical) Harper operator 2 hθ,ϕ = S+S∗ +2 cos(θX +ϕ) restrict to the discrete half-line, that is, acting on ℓ (N). One is thus interested in its bound states (boundaryb states). For that purpose let us first of all choose and bthen fixbE b R in a spectral gap of the two-sided Harper operator h = S + S∗ + 2 cos(θX + ϕ) ∈ θ,ϕ acting on ℓ2(Z), defined with the two-sided shift S. Second of all, let us suppose that the phase is q rational θ =2π p . This is actually not a restriction because one can approximate irrational θ by rational ones and use that T depends continuously on θ so that the signature does not change.

43 The rationality assumption implies that the potential of hθ,ϕ is p-periodic. As the half-sided

Harper operator hθ,ϕ is a Jacobi matrix, its bound states can be calculated by transfer matrix methods, as discussed in detail in [ASV]. The transfer matrix over one period is the following real 2 2 G-unitaryb matrix of unit determinant: × p 1 − E 2 cos(θn + ϕ) 1 E = − − . Tϕ 1 0 n=0 Y 1 E As E is not in the spectrum of hθ,ϕ, the eigenvalues κ, κ− of ϕ are oﬀ the unit circle. This 1 T also assures that T is essentially S -gapped. One now has a bound state for hθ,ϕ if and only if the Dirichlet boundary condition coincides with the contracting direction, namely b 1 1 E = κ , Tϕ 0 0 where κ < 1. For ﬁxed E, this is a trigonometric equation for ϕ. The (weighted) number of its solutions| | is actually equal to the Chern number of the Fermi projection of two-dimensional Harper operator on energies below E [ASV]. This number of solutions can thus take any integer q value (when one is free to vary p as well as E, the only free parameters). For each solution ϕ, one then knows that eıϕ is an eigenvalue of T . A more detailed treatment of this connection can be found in [SV].

A Reminders on Fredholm operators

A bounded operator T B( ) on a Hilbert space is called a Fredholm operator if and only ∈ H if Ran(T ) is closed, dim(Ker(T )) < and dim(Ran(T )⊥) < . Its Fredholm index is then ∞ ∞ deﬁned by Ind(T ) = dim(Ker(T )) dim(Ran(T )⊥). Recall also [Hal] that an operator T B( ) − ∈ H is called bounded from below if there is a constant g > 0 such that Tφ g φ for all φ . Then one can prove the following: T is invertible (as operator, thatk is,k there ≥ k existsk a bounded∈ H inverse) if and only if T is bounded from below and Ran(T ) is dense. Furthermore, T is said to be essentially bounded from below if there exists a constant g > 0 such that T ∗T g 1 except on a ﬁnite dimensional subspace. The following shows that this is connected to Fredholm≥ properties.

Proposition 39 All of the following statements are equivalent:

(i) There exists g > 0 such that T ∗T g 1 except on a ﬁnite dimensional subspace ≥ (ii) 0 σess(T ∗T ) 6∈ (iii) T ∗T is a Fredholm operator

(iv) There is no singular Weyl sequence (φn)n 1 of pairwise orthogonal unit vectors with Tφn 0 ≥ → (v) T is a left semi-Fredholm operator, namely Ran(T ) is closed and dim(Ker(T )) < ∞ (vi) There exists S B( ) such that ST 1 is compact (S is called a left pseudo-inverse) ∈ H − (vii) There exists S B( ) such that T ∗S 1 is compact ∈ H − (viii) T ∗ is a right semi-Fredholm operator, namely Ran(T ∗) is closed and dim(Ran(T ∗)⊥) < ∞ 44 Proof. The equivalence of (i) to (iv) can be found in any text book. Equivalence with (v) is [EE, Corollary I.4.7] and with (vi) [EE, Theorem I.3.13] (actually, it rather follows from the proof therein). Finally equivalence of (vi) with (vii) is [EE, Proposition I.3.9] and with (viii) again the proof of [EE, Theorem I.3.13], combined with [EE, Theorem I.3.7] ✷

Corollary 4 T is a Fredholm operator if and only if T and T ∗ are essentially bounded from below.

Proposition 40 For a Fredholm operator T with Ind(T )=0, one has 0 σ (T ) σ (T ). 6∈ c ∪ r Proof. For a Fredholm operator with vanishing index there exists a compact operator K such that S = T + K is invertible, namely 0 σ(S). Suppose that 0 σ (T ). Then there exists 6∈ ∈ c a singular Weyl sequence (φn)n 1 of pairwise orthogonal unit vectors with Tφn 0. But as ≥ Kφ 0, one also has Sφ 0 and therefore 0 σ(S), which is a contradiction.→ Next n → n → ∈ recall that 0 σ (T ) if and only if T has trivial kernel and Ran(T ) is not dense, namely if ∈ r dim(Ker(T )) = 0 and dim(Ker(T ∗)) > 0. But these two properties are not reconcilable with Ind(T )=0 so that 0 σ (T ). ✷ 6∈ r B Frames, angle spectrum and Fredholm pairs

This appendix resembles a few deﬁnitions and known results about subspaces of a separable Hilbert space . H Deﬁnition 8 Let be a closed subspace of a Hilbert space of dimension k N . Denote 2 2 E 2 H ∈ ∪{∞} ℓ = ℓ ( 1,...,k ). A frame for is an operator Φ: ℓ with = Ran(Φ) and Φ∗Φ = 1. { } E → H E Then Φ⊥ denotes a frame for ⊥. E

If Φ is a frame for , then ΦΦ∗ is obviously the orthogonal projection in on . Note that frames are also partialE isometries. Clearly frames are not unique. The followingH definitionE naturally generalizes finite-dimensional notions. In fact, if φ and ψ are two unit vectors, then 2 2 the angle θ between them is given by cos(θ) = φ∗ψ . | | Definition 9 Let and be two closed subspaces of a Hilbert space with frames Φ and Ψ E F H π respectively. Suppose dim( ) dim( ). Then the angle spectrum σ( , ) [0, 2 ] between and E ≥π F E F ⊂ E is given by the those θ [0, ] the cosine square of which are in the spectrum of (Ψ)∗ΦΦ∗Ψ: F ∈ 2 2 cos (σ( , ) = σ((Ψ)∗ΦΦ∗Ψ) . E F The angle spectrum can be decomposed into point, singular and absolutely continuous spectrum as well as discrete and essential spectrum by using these notions for the self-adjoint operator (Ψ)∗ΦΦ∗Ψ.

45 Because for any operator T one has σ(T ∗T ) 0 = σ(T T ∗) 0 , it follows that σ( , ) π π ∪{ } ∪{ } E F ∪ 2 = σ( , ) 2 in case that dim( ) = dim( ). It is also possible to calculate the angle{ } spectrumF E from∪{ the} associated orthogonalE projections.F For example, if + = , then E F H sin(σ( , )) = σ(ΦΦ∗ ΨΨ∗) R . E F − ∩ ≥ It is possible that the angle spectrum σ( , ) between two closed subspaces is essential at E F 0 so that there are inﬁnitely many almost common directions. Then, even if 0 is not in the point spectrum of σ( , ) so that the intersection of the subspaces is empty as a set, one cannot E F speak of transversal subspaces. The following notion due to Kato [Kat] is to be interpreted as a transversality condition on two subspaces as well as their orthogonal complements up to ﬁnite dimensional intersections.

Definition 10 Two closed subspaces and of a Hilbert space are said to form a Fredholm pair if the dimension of the intersectionE F and the codimensionH of the sum + are finite and, moreover, + is a closed subspace.E∩F Their index is then defined by E F E F ind( , ) = dim( ) codim( + ) . E F E∩F − E F Note that the closedness of + has to be assumed, in general. It does not follow from the closedness of and unless oneE ofF the latter is finite dimensional. Note also that if, say, is finite dimensional,E thenF has to span all but finitely many directions of if and formF a E H E F Fredholm pair. One is, however, mainly interested in the case where both and are infinite dimensional. The above definition coincides with [Fur], but in [ASS] and E (or theF associated E F orthogonal projections) are said to form a Fredholm pair if the dimension of the intersection ⊥ and the codimension of the sum + ⊥ are finite and + ⊥ is closed. In view of the E∩Fconnections to the angles between the subspacesE F presented next,E weF stick to the above choice.

Theorem 11 Let and be two closed subspaces and Φ and Ψ be frames for and . Then the following are equivalent:E F E F (i) and form a Fredholm pair E F (ii) Φ∗Ψ⊥ is a Fredholm operator

(iii) The essential angle spectra σess( , ) and σess( ⊥, ⊥) do not contain 0 E F E F

Proof. Let P = Φ(Φ)∗ be the orthogonal projection on and Q = Ψ⊥(Ψ⊥)∗ the orthogonal E projection on ⊥. It is proved in [ASS, Fur] that (i) is equivalent to the fact that QP seen as an F 2 2 operator from P to Q is a Fredholm operator. But Φ: ℓ P and (Ψ⊥)∗ : Q ℓ are H H → H H → isomorphisms so that QP is Fredholm if and only if (Ψ⊥)∗QP Φ=(Ψ⊥)∗Φ is Fredholm. For the equivalence (ii) (iii) let us recall that an operator T is Fredholm if and only if T ∗T and T T ∗ ⇔ are Fredholm, thus if and only if 0 σess(T ∗T ) and 0 σess(T T ∗). This is applied to T = Φ∗Ψ⊥. 6∈ 6∈ Hence (ii) is equivalent to

0 σess (Ψ⊥)∗ΦΦ∗Ψ⊥ = 1 σess Ψ⊥)∗Φ⊥(Φ⊥)∗Ψ⊥ 0 σess ⊥, ⊥ , 6∈ − ⇐⇒ 6∈ E F and

0 σess Φ∗Ψ⊥(Ψ⊥)∗Φ = 1 σess Φ∗ΨΨ∗Φ 0 σess , ) . 6∈ − ⇐⇒ 6∈ F E 46 But 0 σess , ) is equivalent to 0 σess , ) because these two angle spectra diﬀer at most by the6∈ point Fπ . E 6∈ E F ✷ 2 Next let us recall that if = + is the sum of two closed subspaces with trivial intersection = 0 , then there isH an associatedE F oblique projection (idempotent) deﬁned by Pφ = ψ if E∩F { } φ = ψ + ψ′ with ψ and ψ′ is the unique decomposition. The range of P is , the kernel ∈E ∈F E . The projection P is orthogonal if and only if = ⊥. F E F Proposition 41 Let and be a Fredholm pair and let Φ and Ψ be frames for and . If = + and E= 0 F, then E F H E F E∩F { } 1 P = Φ (Ψ⊥)∗Φ − (Ψ⊥)∗ , (52) is the oblique projection (idempotent) on with range Ran(P )= and kernel Ker(P )= . H E F

Proof. By Theorem 11, the operator (Ψ⊥)∗Φ is Fredholm and therefore Ran((Ψ⊥)∗Φ) is closed 2 so that ℓ = Ran((Ψ⊥)∗Φ) Ker(Φ∗Ψ⊥). Let us show that the kernel of Φ∗Ψ⊥ is trivial. Indeed, if ⊕ Φ∗Ψ⊥v =0 for some vector v, then Ψ⊥v is perpendicular to , thus Ψ⊥v ⊥ ⊥ =( + )⊥ = E ∈E ∩F E2 F 2 0 , the last equality by the hypothesis = + , so that v = 0. Thus (Ψ⊥)∗Φ: ℓ ℓ is { } H E F → surjective. To check the injectivity, suppose that (Ψ⊥)∗Φv =0. Then Φv so that v =0 ∈E∩F by the hypothesis = 0 . Hence (Ψ⊥)∗Φ is bijective and bounded and therefore invertible by the inverse mappingE∩F theorem.{ } ✷

C Riesz projections

The following proposition resembles a few facts about Riesz projections associated to a bounded operator T on a Hilbert space. Proofs can be found, e.g., in [Kat].

Proposition 42 Let ∆ σ(T ) be a separated spectral subset, namely a closed subset which has trivial intersection with the⊂ closure of σ(T ) ∆. Associated to ∆ let Γ be a curve in C σ(T ) with winding number 1 around each point of\∆ and 0 around all points of σ(T ) ∆. The\ Riesz \ projection of T on ∆ is deﬁned as

dz 1 P = (z T )− . (53) ∆ 2πı − IΓ Range and kernel of P are denoted by = Ran(P ) and = Ker(P ). If ∆ = λ is an ∆ E∆ ∆ F∆ ∆ { } isolated point in σ(T ), let us also use the notation Pλ = P∆, λ = ∆ and so on. The following properties hold. E E (i) P is idempotent, namely an oblique projection and and are closed subspaces. Moreover, ∆ E∆ F∆ P∆ is independent of the choice of Γ. 1 (ii) If T is invertible and (Γ)− denotes the path of inversed complex points, one has

dz 1 1 P∆ = (z T − )− . (54) −1 2πı − I(Γ) 47 (iii) If ∆ and ∆′ are disjoint separated spectral subsets, then P∆P∆′ =0 and P∆ ∆′ = P∆ + P∆′ . L ∪ L 1 (iv) For is a disjoint decomposition σ(T )= l=1 ∆l in separated spectral subsets, l=1 P∆l = . (v) is invariant for T and is invariant for T ∗. Moreover, dim( ) = dim( ⊥). E∆ F∆ S E∆ FP∆ (vi) If Φ and Ψ are frames for and ⊥ and Ψ∗ Φ is invertible, then ∆ ∆ E∆ F∆ ∆ ∆ 1 P∆ = Φ∆ Ψ∆∗ Φ∆)− Ψ∆∗ . (vii) The orthogonal projections on and are P P ∗ = Φ Φ∗ and P ∗ P =Ψ Ψ∗ . E∆ F∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ (viii) If dim( ) < , then is the span of the generalized eigenvectors of T to λ. Eλ ∞ Eλ (ix) Let f be an analytic function on the convex closure of σ(T ). Suppose that

f(σ(T ) ∆) f(σ(T ) ∆) = . ∩ ∩ \ ∅

Denote by Qf(∆) the Riesz projection of f(T ) on f(∆), which is a separated spectral subset for f(T ). Then Qf(∆) = P∆.

D Analytic Fredholm theory

This appendix is there to illustrate the diﬃculty linked to the stability of the essential spectrum of non-self-adjoint operators. The following criterion is extracted from the arguments in Section XIII.4 of [RS]. Examples that case (ii) below actually does appear are given in Section 6.4.

Theorem 12 Let T and S be two bounded operators on some separable Hilbert space such that T S is compact. Let C C be one connected component of C σ(T ). Then one of the following two− claims holds true: ⊂ \ (i) C contains a point in the resolvent set of S. (ii) All points of C are eigenvalues of S. In the case (i), the spectrum of S in C is discrete.

1 Proof. Let K = T S. Then z C K(T z)− is analytic and compact-valued. For − ∈ 1 7→ − 1 z C one has S z = (1 K(T z)− )(T z), so the inverse (S z)− exists if and only if ∈ 1 −1 − − − − (1 K(T z)− )− exists. Now one clearly has the dichotomy that either C contains some point − − in the resolvent set of S or it contains none. The ﬁrst case corresponds to (i). Then there exists 1 1 some z0 C such that the inverse (1 K(T z0)− )− exists. But by the analytic Fredholm ∈ − − 1 1 theorem [RS, Theorem VI.16], the inverse (1 K(T z)− )− then exists for all z C except for − − ∈ a discrete set of points. Hence also the spectrum of S lying in C only consists of a discrete set of points. In the second possibility, where no point of C lies in the resolvent set of S, the operator 1 1 K(T z)− is not invertible for any z C. By the Fredholm alternative this implies that for − − ∈ 1 1 each z C there is a vector vz lying in the kernel of 1 K(T z)− . Setting wz =(T z)− vz, one then∈ has (T z)w = Kw , that is Sw = zw . Therefore− − in the second possibility− all points − z z z z z C are eigenvalues of S. ✷ ∈

48 E Spectral ﬂow of paths of essentially gapped unitaries

For the convenience of the reader, some folklore facts about spectral flow are recollected in this appendix. Let Uess( ) be the set of those unitary operators u on a Hilbert space with the H H property that 1 σess(u). Such unitaries will be called essentially gapped. Associated to every 6∈ U (continuous) path γ = (ut)t [t0,t1] in ess( ) one can now define the spectral flow SF(γ) as the ∈ number of eigenvalue crossings through 1,H weighted by the orientation of the crossing. For that purpose, let us first suppose that the number of crossings t [t , t ) 1 σ(u ) is finite { ∈ 0 1 | ∈ t } and does not contain the initial point t0. It is explained below that these transversality and boundary conditions can be considerably relaxed in a straightforward manner. At a crossing ut with 1 σ(u ), let eıθ1,s ,...,eıθl,s be those eigenvalues of the unitary u which are all equal to 1 ∈ t s at s = t. Let us choose them to be continuous in s and call the θk,s ( π, π] also the eigenphases of u . Choose ǫ,δ > 0 such that θ [ δ, δ] for k =1,...,l and s∈ −[t ǫ, t + ǫ] and that there s k,s ∈ − ∈ − are no other eigenphases in [ δ, δ] for s = t and finally θk,s = 0 for those parameters. Let n − 6 6 − and n+ be the number of those of the l eigenphases less than 0 respectively before and after the intersection, and similarly let p and p+ be the number of eigenphases larger than 0 before and − after the crossing. Then the signature of ut is defined by 1 sgn(ut) = (p+ n+ p + n ) = l n+ p . (55) 2 − − − − − − − Note that l sgn(γ ) l and that sgn(γ ) is the effective number of eigenvalues that have − ≤ t ≤ t crossed 1 in the counter-clock sense. Furthermore the signature is stable under perturbations of the path in the following sense: if a crossing is resolved by a perturbation into a series of crossings with lower degeneracy, then the sum of their signatures is equal to sgn(ut). Finally let us remark that, if the phases are differentiable and ∂ θ = 0 for k = 1,...,l, then sgn(u ) is equal to the t k,t 6 t sum of the l signs sgn(∂tθk,t), k = 1,...,l. Yet another equivalent way to calculate sgn(ut) is 1 as the signature of ı (ut)∗∂tut seen as quadratic from on the eigenspace of ut to the eigenvalue 1 (again under the hypothesis that the form is non-degenerate). Now the spectral flow of the path γ is defined by

SF(γ) = sgn(ut) , (56) t X where the sum is over the finite number of points at which 1 σ(u ). If the initial point t is ∈ t 0 on the singular cycle, but say the speeds of the eigenvalues passing through 1 are non-vanishing, the index can still be defined. In order to conserve an obvious concatenation property, only the initial point t0 in (56) is included and not the final point t1 (an alternative would be to give each 1 a weight 2 ). Furthermore, if a path γ is such that 1 σ(ut) for an interval of parameters t, but it is clearly distinguishable how many eigenvalues pass∈ through 1 in the process, then the index can be defined as well. A formal, but obvious definition in these cases is not written out. It is clear from the definition that SF(γ) is a homotopy invariant under homotopies keeping the end points of γ fixed. Acknowledgements: The author profited from several discussions with Stephane Merigon and received financial support by the DFG. This work has a follow up [SV] which analyzes homotopy invariants for essentially gapped operators with a supplementary symmetry and also deals with unbounded J-isometries.

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